// Numbas version: finer_feedback_settings
{"name": "MATH6005 Vectors", "feedback": {"showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "allQuestions": true, "shuffleQuestions": false, "percentPass": "0", "duration": 0, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": false}, "metadata": {"description": "<p>Questions on vector arithmetic and vector operations, including dot and cross product, as well as the vector equations of planes and lines.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Angle between vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "showCorrectAnswer": true, "minValue": "{angle}", "maxValue": "{angle}", "precision": "precision", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1}], "type": "gapfill", "prompt": "<p>Find the angle between $\\boldsymbol{v}$ and $\\boldsymbol{w}$, in radians.</p>\n<p>Note the angle must be in the range $0$&nbsp;to $\\pi$.</p>\n<p>Give your answer to {precision} decimal places.</p>\n<p>Angle in radians&nbsp;= [[0]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variables": {"s1": {"templateType": "anything", "group": "Initial vectors", "definition": "random(1,-1)", "description": "", "name": "s1"}, "w": {"templateType": "anything", "group": "Initial vectors", "definition": "direction_w[0]*s3 + direction_w[1]*s4", "description": "", "name": "w"}, "v": {"templateType": "anything", "group": "Initial vectors", "definition": "direction_v[0]*s1 + direction_v[1]*s2", "description": "", "name": "v"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=3,2,3)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,2,1)", "description": "", "name": "c"}, "s4": {"templateType": "anything", "group": "Initial vectors", "definition": "if(s1=s3 ,-s2,random(-1,1))", "description": "", "name": "s4"}, "direction_v": {"templateType": "anything", "group": "Initial vectors", "definition": "map(units[j],j,shuffle(0..2))", "description": "", "name": "direction_v"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,3)", "description": "", "name": "t"}, "direction_w": {"templateType": "anything", "group": "Initial vectors", "definition": "map(units[j],j,shuffle(0..2))", "description": "", "name": "direction_w"}, "tb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=3,0,s4)", "description": "", "name": "tb"}, "fa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,0,s1)", "description": "", "name": "fa"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2,3)", "description": "", "name": "u"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,2,1)", "description": "", "name": "a"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{fa*fb+sa*sb+ta*tb}", "description": "", "name": "g"}, "fb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,0,s3)", "description": "", "name": "fb"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=3,2,3)", "description": "", "name": "d"}, "units": {"templateType": "anything", "group": "Initial vectors", "definition": "map(vector(x),x,list(id(3)))", "description": "", "name": "units"}, "sb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=2,0,if(u=1,s3,s4))", "description": "", "name": "sb"}, "precision": {"templateType": "anything", "group": "Result", "definition": "3", "description": "", "name": "precision"}, "ta": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=3,0,s2)", "description": "", "name": "ta"}, "s3": {"templateType": "anything", "group": "Initial vectors", "definition": "random(1,-1)", "description": "", "name": "s3"}, "angle": {"templateType": "anything", "group": "Result", "definition": "arccos(dot(v,w)/(len(v)*len(w)))", "description": "", "name": "angle"}, "sa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=2,0,if(t=1,s1,s2))", "description": "", "name": "sa"}, "s2": {"templateType": "anything", "group": "Initial vectors", "definition": "random(1,-1)", "description": "", "name": "s2"}}, "ungrouped_variables": ["t", "u", "a", "b", "c", "d", "fa", "fb", "g", "sa", "sb", "ta", "tb"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [{"variables": ["s1", "s2", "s3", "s4", "units", "direction_v", "direction_w", "v", "w"], "name": "Initial vectors"}, {"variables": ["angle", "precision"], "name": "Result"}], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are given the vectors $\\boldsymbol{v} = \\var{v}$, $\\boldsymbol{w} = \\var{w}$ in $\\mathbb{R}^3$.</p>", "tags": ["angle between vectors", "angle beween two vectors", "checked2015", "degrees and radians", "dot product", "finding the angle between vectors", "inner product", "MAS1602", "mas1602", "radians", "scalar product", "vectors"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2015</strong></p>\n<p>Added tags</p>\n<p><strong></strong></p>\n<p><strong>16/07/2012:<br /></strong></p>\n<p>Added tags.</p>\n<p>Question appears to be working correctly.<strong><br /></strong></p>\n<p><strong>Moved the \\rightarrow to the correct place in the solution.<br /></strong></p>\n<p><strong>&nbsp;</strong></p>\n<p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given vectors&nbsp;&nbsp;$\\boldsymbol{v,\\;w}$, find the angle between them.</p>"}, "advice": "<p>Use the formula, $\\boldsymbol{v \\cdot w} = \\lVert \\boldsymbol{v} \\rVert \\lVert \\boldsymbol{w} \\rVert \\cos(\\theta)$m where $\\theta$ is the angle between the vectors.</p>\n<p>Here</p>\n<p>\\begin{align}<br/>\\lVert \\boldsymbol{v} \\rVert &amp;= \\simplify[]{sqrt({s1}^2 + {s2}^2)} \\\\<br/>&amp;= \\sqrt{2}, \\\\[1em]<br/>\\lVert \\boldsymbol{w} \\rVert &amp;= \\simplify[]{sqrt({s3}^2 + {s4}^2)} \\\\<br/>&amp;= \\sqrt{2}, \\\\[1em]<br/>\\boldsymbol{v \\cdot w} &amp;= \\var{v} \\boldsymbol{\\cdot} \\var{w} \\\\<br/>&amp;= \\var{dot(v,w)}<br/>\\end{align}</p>\n<p>So</p>\n<p>\\begin{align}<br/>\\cos(\\theta) &amp;= \\frac{\\var{dot(v,w)}}{\\sqrt{2}\\sqrt{2}} = \\simplify[std]{{dot(v,w)}/2} \\\\<br/> \\implies \\theta &amp;= \\arccos\\left(\\simplify[std]{{dot(v,w)}/{2}}\\right) \\\\ <br/>&amp;= \\var{precround(angle,precision)} \\text{ radians}<br/>\\end{align}</p>"}, {"name": "Determine if vectors are perpendicular", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(c,d,f)", "description": "", "name": "w"}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(a,b,lambda)", "description": "", "name": "v"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,4,5,10)", "description": "", "name": "f"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "lambda": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(-a*c-b*d)/f", "description": "", "name": "lambda"}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s5"}, "mu1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "lcm(random(-5..5 except 0),f)", "description": "", "name": "mu1"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "mu2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "lcm(random(-5..5 except 0),f)", "description": "", "name": "mu2"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "d"}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "mu1*v+mu2*w", "description": "", "name": "u"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "lambda", "mu1", "mu2", "v", "w", "u"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\"><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">When are vectors $\\boldsymbol{v,\\;w}$ orthogonal</span></span><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">?</span></p>\n<p><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">Part b) is not answered in Advice, the given solution is for a different question.</span></p>"}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Find $\\lambda \\in \\mathbb{R}$ such that $\\boldsymbol{v}$ and $\\boldsymbol{w}$ are orthogonal.</p>\n<p>$\\lambda = $&nbsp;[[0]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "lambda", "maxValue": "lambda", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "variableReplacements": [], "marks": 1.5, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Find $\\lambda \\in \\mathbb{R}$ such that the vector $\\boldsymbol{u} = \\simplify[fractionnumbers]{{u}}$ is contained in the&nbsp;plane through the origin parallel to $\\boldsymbol{v}$ and $\\boldsymbol{w}$.</p>\n<p>$\\lambda =$ [[0]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "lambda", "maxValue": "lambda", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "variableReplacements": [], "marks": 0.5, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "statement": "<p>You are given the vectors $\\boldsymbol{v} = \\begin{pmatrix}\\var{a} \\\\ \\var{b} \\\\ \\lambda \\end{pmatrix}$ and $\\boldsymbol{w} = \\begin{pmatrix} \\var{c} \\\\ \\var{d} \\\\ \\var{f} \\end{pmatrix}$.</p>\n<p><em>Enter your answers to the following questions as fractions or integers, not decimals.</em></p>", "tags": ["checked2015", "dot product", "finding perpendicular vectors", "inner product", "perpendicular vectors", "product", "scalar product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "u<>vector(0,0,0)", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>$\\boldsymbol{v}$ and $\\boldsymbol{w}$ are perpendicular to one another when $\\boldsymbol{v} \\cdot \\boldsymbol{w} = 0$.</p>\n<p>Now</p>\n<p>\\begin{align}<br/>\\boldsymbol{v} \\cdot \\boldsymbol{w} &amp;= \\simplify[]{{a}*{c}+{b}*{d}+lambda*{f}} \\\\ <br/>&amp;= \\simplify[std]{{f}*lambda+{a*c+b*d}}<br/>\\end{align}</p>\n<p>Hence</p>\n<p>\\[\\boldsymbol{v} \\cdot \\boldsymbol{w} = 0 \\implies \\simplify[std]{{f}*lambda+{a*c+b*d}}=0 \\implies \\lambda = \\simplify[std]{{-a*c-b*d}/{f}}\\]</p>\n<h4>b)</h4>\n<p>$\\boldsymbol{v}$ is in the $xy$ plane when $\\lambda=0$.</p>"}, {"name": "Dot and cross product combinations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"displayType": "radiogroup", "layout": {"type": "all", "expression": ""}, "choices": ["$\\boldsymbol{(v\\cdot w)\\cdot u}$", "$\\boldsymbol{(v\\cdot w)u}$", "$\\boldsymbol{(v\\cdot w)\\times u}$", "$\\boldsymbol{(v\\times w)\\times u}$", "<p>$\\boldsymbol{(v\\times w)\\cdot u}$</p>"], "variableReplacementStrategy": "originalfirst", "matrix": [[0, 0, 0.4], [0, 0.4, 0], [0, 0, 0.4], [0, 0.4, 0], [0.4, 0, 0]], "shuffleChoices": true, "type": "m_n_x", "maxAnswers": 0, "marks": 0, "warningType": "none", "scripts": {}, "minMarks": 0, "minAnswers": 0, "maxMarks": 0, "shuffleAnswers": true, "showCorrectAnswer": true, "variableReplacements": [], "answers": ["<p>Scalar</p>", "<p>Vector</p>", "<p>Undefined</p>"]}], "variables": {}, "ungrouped_variables": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Given the vectors $\\boldsymbol{v}$, $\\boldsymbol{w}$, $\\boldsymbol{u}$ in $\\mathbb{R}^3$, state whether the following quantities are scalars (real numbers), vectors (elements of $\\mathbb{R}^3$) or undefined.</p>\n<p>In this question, the&nbsp;symbol $\\cdot$ denotes the inner product and $\\times$ always denotes the cross product.</p>", "tags": ["checked2015", "cross product", "dot product", "inner product", "MAS1602", "mas1602", "scalar product", "scalars", "unused", "vector", "Vector", "vector product", "vectors"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n        \t\t<p><strong>15/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p><strong>16/07/2012:</strong></p>\n        \t\t<p><strong>Added tags.<br /></strong></p>\n        \t\t<p><strong>&nbsp;</strong></p>\n        \t\t<p><strong>&nbsp;</strong></p>\n        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Determine if various combinations of vectors are defined or not.</p>"}, "advice": "<p>1. $\\boldsymbol{(v\\cdot w)\\cdot u}$ is <b>undefined</b> as $\\boldsymbol{v\\cdot w}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{u}$.</p>\n<p>2. $\\boldsymbol{(v\\cdot w) u}$ is a <b>vector</b> and is a scalar multiple of $\\boldsymbol{u}$ as $\\boldsymbol{v \\cdot w}$ is a scalar.</p>\n<p>3. $\\boldsymbol{(v \\cdot w)\\times u}$ is <b>undefined</b> as $\\boldsymbol{v\\cdot w}$ is a scalar and the cross product is only defined between vectors.</p>\n<p>4. $\\boldsymbol{(v\\times w)\\times u}$ is a <b>vector</b> as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the cross product between vectors produces a vector.</p>\n<p>5. $\\boldsymbol{(v\\times w)\\cdot u}$ is a <b>scalar</b> as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the inner or dot product is between vectors and produces a scalar.</p>"}, {"name": "Inner product of two vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "{inner}", "maxValue": "{inner}", "variableReplacements": [], "marks": 2}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Find $\\boldsymbol{v&nbsp;\\cdot&nbsp;w} = $&nbsp;[[0]]</p>", "variableReplacements": [], "marks": 0}], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s5", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "name": "c", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "f", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s4", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}, "inner": {"group": "Ungrouped variables", "templateType": "anything", "definition": "{a*c+b*d+f*g}", "name": "inner", "description": ""}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s3", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "a", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "name": "d", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "g", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "inner"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "variable_groups": [], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are given the vectors $\\boldsymbol{v}= \\var{vector(a,b,g)}$ and $\\boldsymbol{w} = \\var{vector(c,d,f)}$ in $\\mathbb{R}^3$.</p>", "tags": ["checked2015", "dot product", "dot product of two vectors", "inner product", "mas1602", "MAS1602", "scalar product", "three dimensional vectors", "unused", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n            \t\t<p><strong>15/07/2012:</strong></p>\n            \t\t<p>Added tags.</p>\n            \t\t<p><strong>16/07/2012:</strong></p>\n            \t\t<p>Added tags.<strong><br /></strong></p>\n            \t\t<p>Question appears to be working correctly.</p>\n            \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given vectors $\\boldsymbol{v}$ and $\\boldsymbol{w}$, find their inner product.</p>"}, "advice": "<p>\\begin{align}<br/>\\boldsymbol{v \\cdot w} &amp;= \\var{vector(a,b,g)} \\boldsymbol{\\cdot} \\var{vector(c,d,f)} \\\\<br/>&amp;= \\simplify[]{{a}*{c}+{b}*{d}+{g}*{f}} \\\\<br/> &amp;= \\var{inner}<br/>\\end{align}</p>"}, {"name": "Vector cross product", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "result": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(vector(a,b,g),vector(c,d,f))", "description": "", "name": "result"}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s5"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "f"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "d"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "result", "s1", "s2", "s3", "s4", "s5"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>Find</p>\n<p>$\\boldsymbol{v} \\times \\boldsymbol{w} = $&nbsp;[[0]]</p>", "scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "result", "showCorrectAnswer": true, "allowResize": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "3"}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are given the vectors $\\boldsymbol{v} = \\var{vector(a,b,g)}$, $\\boldsymbol{w} = \\var{vector(c,d,f)}$.</p>", "tags": ["3 dimensional vector", "checked2015", "cross product", "three dimensional vectors", "unused", "Vector", "vector", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>14/7/2015</strong></p>\n<p>Added unused tag</p>\n<p><strong></strong></p>\n<p><strong>16/07/2012:</strong></p>\n<p><strong>Added tags.<br /></strong></p>\n<p><strong>Question appears to be working correctly.<br /></strong></p>\n<p><strong>&nbsp;</strong></p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given vectors $\\boldsymbol{A,\\;B}$, find $\\boldsymbol{A\\times B}$</p>"}, "advice": "<p>\\begin{align}<br/>\\boldsymbol{v} \\times \\boldsymbol{w} &amp;= \\begin{pmatrix} \\simplify[basic]{{b}*{f}-{g}*{d}} \\\\ \\simplify[basic]{{g}*{c}-{a}*{f}} \\\\ \\simplify[basic]{{a}*{d}-{b}*{c}} &nbsp;\\end{pmatrix} \\\\[1em]<br/>&amp;= \\var{result}<br/>\\end{align}</p>"}, {"name": "Angle between two vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "tags": ["angle between vectors", "angle beween two vectors", "degrees and radians", "dot product", "finding the angle between vectors", "inner product", "radians", "scalar product", "vectors"], "type": "question", "advice": "<p>Bruker formelen:</p>\n  <p>$\\boldsymbol{A \\cdot B} = |\\boldsymbol{A}||\\boldsymbol{B}|\\cos(\\theta)$ der $\\theta$ er vinkelen mellom vektorene.</p>\n  <p>Her er $|\\boldsymbol{A}| = \\sqrt{ (\\var{s1})^2+(\\var{s2})^2} = \\simplify[all]{sqrt({s1^2+s2^2})},\\;\\;\\;|\\boldsymbol{B}| = \\sqrt{ (\\var{s3})^2+(\\var{s4})^2} = \\simplify[all]{sqrt({s3^2+s4^2})}$</p>\n  <p>og</p>\n  <p>$\\boldsymbol{A \\cdot B} = (\\var{fa},\\var{sa}, \\var{ta}) \\cdot (\\var{fb},\\var{sb}, \\var{tb}) = \\var{g}$.</p>\n  <p>Slik at \\[\\begin{eqnarray*} \\cos(\\theta)&amp;=&amp;\\frac{\\var{g}}{\\sqrt{2}\\sqrt{2}} = \\simplify[std]{{g}/{2}}\\\\ \\Rightarrow \\theta &amp;=&amp;\\arccos\\left(\\simplify[std]{{g}/{2}}\\right)\\\\ &amp;=&amp;\\var{angle}\\,^{\\circ} \\end{eqnarray*} \\]<br/>Konvertering fra grader til radianer gj&oslash;res ved &aring; multiplisere vinkel i grader med $\\displaystyle \\frac{\\pi}{180}$.</p>\n  <p>Da blir $\\displaystyle \\var{angle}\\,^{\\circ}=\\simplify[std]{({angle}*pi)/{180}= {precround(angle*pi/180,4)}}$ radianer i 4 siffers n&oslash;yaktighet.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Angle in degrees = [[0]]$^{\\circ}$</p>\n<p>Angle in radians = [[1]]radians.</p>\n<p>Note that you can input the radians as a decimal to 4 decimal places or as a mulptiple of $\\pi$. You input $\\pi$ as pi.</p>", "gaps": [{"minvalue": "{angle}", "type": "numberentry", "maxvalue": "{angle}", "marks": 1.0, "showPrecisionHint": false}, {"expectedvariablenames": [], "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "showpreview": true, "vsetrangepoints": 5.0, "checkingtype": "absdiff", "marks": 1.0, "answer": "{precround(angle*pi/180,4)}", "checkvariablenames": false, "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "<p>Given the vectors<br/>$\\mathbf{A}=\\var{fa}\\mathbf{i}+\\var{sa}\\mathbf{j}+\\var{ta}\\mathbf{k},\\;\\;\\;\\boldsymbol{B}=\\var{fb}\\mathbf{i}+\\var{sb}\\mathbf{j}+ \\var{tb}\\mathbf{k}$</p>\n<p>Find the angle between these vectors in degrees and radians.</p>\n<p>Note that the angle must be between $0\\,^{\\circ}$ and $180\\,^{\\circ}$ (between $0$ and $\\pi$ radians)</p>", "variable_groups": [], "progress": "ready", "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "if(t=1,2,1)", "name": "a"}, "c": {"definition": "if(u=1,2,1)", "name": "c"}, "b": {"definition": "if(t=3,2,3)", "name": "b"}, "angle": {"definition": "precround(180/pi*arccos(g/2),1)", "name": "angle"}, "d": {"definition": "if(u=3,2,3)", "name": "d"}, "g": {"definition": "{fa*fb+sa*sb+ta*tb}", "name": "g"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "s4": {"definition": "if(s1=s3 ,-s2,random(-1,1))", "name": "s4"}, "fa": {"definition": "if(t=1,0,s1)", "name": "fa"}, "fb": {"definition": "if(u=1,0,s3)", "name": "fb"}, "u": {"definition": "random(1,2,3)", "name": "u"}, "t": {"definition": "random(1,2,3)", "name": "t"}, "sb": {"definition": "if(u=2,0,if(u=1,s3,s4))", "name": "sb"}, "sa": {"definition": "if(t=2,0,if(t=1,s1,s2))", "name": "sa"}, "tb": {"definition": "if(u=3,0,s4)", "name": "tb"}, "ta": {"definition": "if(t=3,0,s2)", "name": "ta"}}, "metadata": {"notes": "\n        \t\t<p><strong>15/07/2012:</strong></p>\n        \t\t    <p>Added tags.</p>\n        \t\t    <p><strong>16/07/2012:<br /></strong></p>\n        \t\t    <p>Added tags.</p>\n        \t\t    <p>Question appears to be working correctly.<strong><br /></strong></p>\n        \t\t    <p>Moved the \\rightarrow to the correct place in the solution.<strong><br /></strong></p>\n        \t\t    <p><strong>&nbsp;</strong></p>\n        \t\t    <p>&nbsp;</p>\n        \t\t", "description": "<p>Gitt vektorene&nbsp; $\\boldsymbol{A,\\;B}$, finn vinkelen mellom dem.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Cross product 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "inner"], "tags": ["3 dimensional vector", "cross product", "Rebel", "REBEL", "rebel", "rebelmaths", "three dimensional vectors", "Vector", "vector", "vector product", "vectors"], "advice": "<p>\\[ \\begin{eqnarray*} \\boldsymbol{a\\times b}&amp;=&amp; \\begin{vmatrix} \\boldsymbol{i} &amp; \\boldsymbol{j} &amp;\\boldsymbol{k}\\\\ \\var{a} &amp; \\var{b} &amp; \\var{g}\\\\ \\var{c} &amp; \\var{d} &amp; \\var{f} \\end{vmatrix}\\\\ \\\\ &amp;=&amp;\\simplify[]{({b}*{f}-{g}*{d})v:i + ({g}*{c} - {a}*{f})v:j+({a}*{d}-{b}*{c})v:k}\\\\ \\\\ &amp;=&amp;\\simplify[std]{{b*f-g*d}v:i+{g*c-a*f}v:j+{a*d-b*c}v:k} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Find $\\boldsymbol{a\\times b} =\\;\\;$ [[0]]$\\boldsymbol{i}$+[[1]]$\\boldsymbol{j}$+[[2]]$\\boldsymbol{k}$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{b*f-g*d}", "minValue": "{b*f-g*d}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 6, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{g*c-a*f}", "minValue": "{g*c-a*f}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 6, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{a*d-b*c}", "minValue": "{a*d-b*c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 6, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Given the vectors:<br/>\\[\\boldsymbol{a}=\\simplify[std]{{a}v:i+{b}v:j+{g}v:k},\\;\\;\\;\\boldsymbol{b}=\\simplify[std]{{c}v:i+{d}v:j+{f}v:k}\\]</p>\n<p>answer the following question:</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s2*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "s4*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "inner": {"definition": "{a*c+b*d+f*g}", "templateType": "anything", "group": "Ungrouped variables", "name": "inner", "description": ""}}, "metadata": {"description": "<p>Given vectors $\\boldsymbol{a,\\;b}$, find $\\boldsymbol{a\\times b}$</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "resultant displacement", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": ["rebel", "REBEL", "Rebel", "rebelmaths", "resultant displacement"], "metadata": {"description": "<p>resultant displacement</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(10..90)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(10..90)", "description": "", "templateType": "anything", "can_override": false}, "name1": {"name": "name1", "group": "Ungrouped variables", "definition": "random('John',  'Michael', 'Keane', 'Peter')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "name1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{name1} travels {a} km east and then {b} km south. Determine how far {name1} is from his starting point and give his bearings with respect to the eastward direction.</p>\n<p>Distance [[0]] km (to the nearest km)</p>\n<p>Angle: East [[1]] $^\\circ$ South (to the nearest degree)</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 10, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "sqrt(a^2+b^2)", "maxValue": "sqrt(a^2+b^2)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 10, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "arctan(b/a)/pi*180", "maxValue": "arctan(b/a)/pi*180", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": 0, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Vectors: when perpendicular 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4"], "tags": ["dot product", "finding perpendicular vectors", "inner product", "perpendicular vectors", "product", "scalar product", "vectors"], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>$\\boldsymbol{A}$ and $\\boldsymbol{B}$ are perpendicular to one another when $\\boldsymbol{A \\cdot B} = 0$.</p>\n<p>Now \\[ \\begin{eqnarray*}\\boldsymbol{A \\cdot B} &amp;=&amp; \\simplify[]{{a}*{c}+{b}*{d}+lambda*{f}}\\\\ &amp;=&amp; \\simplify[std]{{f}*lambda+{a*c+b*d}} \\end{eqnarray*} \\]<br/>Hence \\[\\boldsymbol{A \\cdot B} = 0 \\Rightarrow \\simplify[std]{{f}*lambda+{a*c+b*d}=0} \\Rightarrow \\lambda = \\simplify[std]{{-a*c-b*d}/{f}}\\]</p>\n<h4>b)</h4>\n<p>$\\boldsymbol{A}$ is in the $xy$ plane when $\\lambda=0$.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Find $\\lambda$ such that $\\boldsymbol{A}$ and $\\boldsymbol{B}$ are perpendicular to one another:</p>\n<p>$\\lambda =\\;\\;$ [[0]].</p>\n<p>Enter your answer as a fraction or an integer and not as a decimal.</p>", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Enter as a fraction or an integer and not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1.5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{-a*c-b*d}/{f}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "<p>Find $\\lambda$ such that $\\boldsymbol{A}$ is in the $xy$ plane:</p>\n<p>$\\lambda =\\;\\;$ [[0]].</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 0.5, "maxValue": "{0}", "minValue": "{0}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "<p>Given the vectors:<br/>\\[\\boldsymbol{A}=\\simplify[std]{{a}v:e_1+{b}v:e_2+lambda*v:e_3},\\;\\;\\;\\boldsymbol{B}=\\simplify[std]{{c}v:e_1+{d}v:e_2+{f}v:e_3}\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s2*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "s4*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "s1*random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2,4,5,10)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>15/07/2012:</strong></p>\n        \t\t<p>Added tags.</p>\n        \t\t<p>Last part is too easy.</p>\n        \t\t<p><strong>16/07/2012:</strong></p>\n        \t\t<p><strong>Added tags.</strong></p>\n        \t\t<p><strong>Question appears to be working correctly.<br /></strong></p>\n        \t\t<p><strong>Agree that last part is too easy.<br /></strong></p>\n        \t\t", "description": "<p><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\"><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">When are vectors $\\boldsymbol{A,\\;B}$&nbsp;</span></span><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">perpendicular?</span></p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Work done by a force", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["p2", "f1", "f2", "f3", "p3", "p1", "q3", "q1", "q2"], "tags": ["rebel", "Rebel", "REBEL", "rebelmaths", "Vector", "vector", "work"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "<p>Find the work done by a force $\\mathbf{F}=$ {f1}$\\mathbf{i}+$ {f2}$\\mathbf{j}+${f3}$\\mathbf{k}$ in moving an object from the point P({p1},{p2},{p3}) to the point Q({q1},{q2},{q3}).</p>\n<p>[[0]] Joules</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>First find the displacement vector $\\simplify{vec:d}$= $\\simplify{vec:PQ}$.&nbsp;</p>\n<p>Note</p>\n<p>$\\simplify{vec:d =vec:PQ =vec:Q-vec:P}$.&nbsp;</p>\n<p>Work done$=$ $\\simplify{vec:F}$ $\\cdot$ $\\simplify{vec:d}$ &nbsp;(Scalar Product)</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "f1*(q1-p1)+f2*(q2-p2)+f3*(q3-p3)", "minValue": "f1*(q1-p1)+f2*(q2-p2)+f3*(q3-p3)", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 20, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"p2": {"definition": "random(-9..2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "p2", "description": ""}, "f1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "f3": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}, "q2": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "q2", "description": ""}, "p3": {"definition": "random(-9..2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "p3", "description": ""}, "q3": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "q3", "description": ""}, "q1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "q1", "description": ""}, "p1": {"definition": "random(-9..2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}}, "metadata": {"description": "<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Trigonometry and vectors: Find angles to nearest degree", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/undefined_YwBJcjH", "/srv/numbas/media/question-resources/undefined_YwBJcjH"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["epphys", "jsxgraph", "Jsxgraph", "JSXgraph", "Triangle", "triangle", "trig", "trigonometry", "vectors"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"anglelist": {"name": "anglelist", "group": "Ungrouped variables", "definition": "random(['$\\\\theta$',3],['$\\\\phi$',4])", "description": "", "templateType": "anything", "can_override": false}, "scale": {"name": "scale", "group": "Ungrouped variables", "definition": "24/triples[1]", "description": "", "templateType": "anything", "can_override": false}, "triples": {"name": "triples", "group": "Ungrouped variables", "definition": "random([[3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [16, 63, 65], [20, 21, 29], [20, 99, 101], [24, 143, 145], [28, 45, 53], [33, 56, 65], [36, 77, 85], [39, 80, 89], [44, 117, 125], [48, 55, 73], [51, 140, 149], [52, 165, 173], [57, 176, 185], [60, 91, 109], [60, 221, 229], [65, 72, 97], [68, 285, 293], [69, 260, 269], [75, 308, 317], [76, 357, 365], [84, 187, 205], [84, 437, 445], [85, 132, 157], [87, 416, 425], [88, 105, 137], [92, 525, 533], [93, 476, 485], [95, 168, 193], [96, 247, 265], [100, 621, 629], [104, 153, 185], [105, 208, 233], [105, 608, 617], [111, 680, 689], [115, 252, 277], [119, 120, 169], [120, 209, 241], [120, 391, 409], [132, 475, 493], [133, 156, 205], [135, 352, 377], [136, 273, 305], [140, 171, 221], [145, 408, 433], [152, 345, 377], [155, 468, 493], [156, 667, 685], [160, 231, 281], [161, 240, 289], [165, 532, 557], [168, 425, 457], [168, 775, 793], [175, 288, 337], [180, 299, 349], [184, 513, 545], [185, 672, 697], [189, 340, 389], [195, 748, 773], [200, 609, 641], [203, 396, 445], [204, 253, 325], [205, 828, 853], [207, 224, 305], [215, 912, 937], [216, 713, 745], [217, 456, 505], [220, 459, 509], [225, 272, 353], [228, 325, 397], [231, 520, 569], [232, 825, 857], [240, 551, 601], [248, 945, 977], [252, 275, 373], [259, 660, 709], [260, 651, 701], [261, 380, 461], [273, 736, 785], [276, 493, 565], [279, 440, 521], [280, 351, 449], [280, 759, 809], [287, 816, 865], [297, 304, 425], [300, 589, 661], [301, 900, 949], [308, 435, 533], [315, 572, 653], [319, 360, 481], [333, 644, 725], [336, 377, 505], [336, 527, 625], [341, 420, 541], [348, 805, 877], [364, 627, 725], [368, 465, 593], [369, 800, 881], [372, 925, 997], [385, 552, 673], [387, 884, 965], [396, 403, 565], [400, 561, 689], [407, 624, 745], [420, 851, 949], [429, 460, 629], [429, 700, 821], [432, 665, 793], [451, 780, 901], [455, 528, 697], [464, 777, 905], [468, 595, 757], [473, 864, 985], [481, 600, 769], [504, 703, 865], [533, 756, 925], [540, 629, 829], [555, 572, 797], [580, 741, 941], [615, 728, 953], [616, 663, 905], [696, 697, 985]])\n", "description": "<p>Some the following were too skinny and so were removed.<br /><br /><br />random([[3,4,5], [5,12,13], [7,24,25], [8,15,17], [9,40,41],<br />[11,60,61], [12,35,37], [13,84,85], [15,112,113], [16,63,65],<br /> [19,180,181], [20,21,29], [20,99,101],<br />[23,264,265], [24,143,145], [25,312,313], [27,364,365], [28,45,53],<br />[28,195,197], [31,480,481], [32,255,257], [33,56,65],<br />[33,544,545], [35,612,613], [36,77,85], [36,323,325], [37,684,685],<br />[39,80,89], [40,399,401], [41,840,841], [43,924,925],<br />[44,117,125], [44,483,485], [48,55,73], [48,575,577], [51,140,149],<br />[52,165,173], [52,675,677], [56,783,785], [57,176,185], [60,91,109],<br />[60,221,229], [60,899,901], [65,72,97], [68,285,293], [69,260,269],<br />[75,308,317], [76,357,365], [84,187,205], [84,437,445], [85,132,157],<br />[87,416,425], [88,105,137], [92,525,533], [93,476,485], [95,168,193],<br />[96,247,265], [100,621,629], [104,153,185], [105,208,233], [105,608,617],<br />[108,725,733], [111,680,689], [115,252,277], [116,837,845], [119,120,169],<br />[120,209,241], [120,391,409], [123,836,845], [129,920,929],<br />[132,475,493], [133,156,205], [135,352,377], [136,273,305], [140,171,221],<br />[145,408,433], [152,345,377], [155,468,493], [156,667,685], [160,231,281],<br />[161,240,289], [165,532,557], [168,425,457], [168,775,793], [175,288,337],<br />[180,299,349], [184,513,545], [185,672,697], [189,340,389], [195,748,773],<br />[200,609,641], [203,396,445], [204,253,325], [205,828,853], [207,224,305],<br />[215,912,937], [216,713,745], [217,456,505], [220,459,509], [225,272,353],<br />[228,325,397], [231,520,569], [232,825,857], [240,551,601], [248,945,977],<br />[252,275,373], [259,660,709], [260,651,701], [261,380,461], [273,736,785],<br />[276,493,565], [279,440,521], [280,351,449], [280,759,809], [287,816,865],<br />[297,304,425], [300,589,661], [301,900,949], [308,435,533], [315,572,653],<br />[319,360,481], [333,644,725], [336,377,505], [336,527,625], [341,420,541],<br />[348,805,877], [364,627,725], [368,465,593], [369,800,881], [372,925,997],<br />[385,552,673], [387,884,965], [396,403,565], [400,561,689], [407,624,745],<br />[420,851,949], [429,460,629], [429,700,821], [432,665,793], [451,780,901],<br />[455,528,697], [464,777,905], [468,595,757], [473,864,985], [481,600,769],<br />[504,703,865], [533,756,925], [540,629,829], [555,572,797], [580,741,941],<br />[615,728,953], [616,663,905], [696,697,985]])</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "if(vec1[2]=1,triples[2],'')", "description": "<p>use&nbsp;https://www.mathsisfun.com/numbers/pythagorean-triples.html</p>\n<p></p>\n<p>so always integer and we could scale by k for more randomness.</p>", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(vec1[0]=1,triples[0],'')", "description": "", "templateType": "anything", "can_override": false}, "deltax": {"name": "deltax", "group": "Ungrouped variables", "definition": "precround(-12+2*triples[1]/triples[0],4)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "if(vec1[1]=1,triples[1],'')", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "ansrad*180/(pi)", "description": "", "templateType": "anything", "can_override": false}, "component": {"name": "component", "group": "Ungrouped variables", "definition": "if(vec1=[1,1,0], [[\"vertical\",v], [\"horizontal\",h]],\nif(vec1=[1,0,1], [[\"vertical\",v], [\"total\",d]],\nif(vec1=[0,1,1], [[\"horizontal\",h],[\"total\",d]],'error')))", "description": "", "templateType": "anything", "can_override": false}, "name": {"name": "name", "group": "Ungrouped variables", "definition": "random([\"Ben\", \"he\"], [\"Annie\", \"she\"], [\"Matt\", \"he\"], [\"David\", \"he\"], [\"Steve\", \"he\"], [\"David\", \"he\"], [\"Scott\", \"he\"], [\"Fran\", \"she\"], [\"Jenny\", \"she\"], [\"Lyn\", \"she\"], [\"Judy-anne\", \"she\"], [\"Courtney\", \"she\"])", "description": "", "templateType": "anything", "can_override": false}, "ansrad": {"name": "ansrad", "group": "Ungrouped variables", "definition": "if(anglelist[1]=3,if(vec1[0]=0, arccos(h/d),if(vec1[1]=0,arcsin(v/d),arctan(v/h))),\nif(vec1[0]=0, arcsin(h/d),if(vec1[1]=0,arccos(v/d),arctan(h/v))))", "description": "", "templateType": "anything", "can_override": false}, "vec1": {"name": "vec1", "group": "Ungrouped variables", "definition": "shuffle([1,1,0])", "description": "", "templateType": "anything", "can_override": false}, "angle": {"name": "angle", "group": "Ungrouped variables", "definition": "anglelist[0]", "description": "", "templateType": "anything", "can_override": false}, "tritop": {"name": "tritop", "group": "Ungrouped variables", "definition": "precround(triples[0]*scale,4)/2", "description": "<p>top of triangle for jsxgraph, keeping same ratios.</p>", "templateType": "anything", "can_override": false}, "switcharoo": {"name": "switcharoo", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["triples", "vec1", "v", "h", "d", "scale", "tritop", "deltax", "switcharoo", "ansrad", "anglelist", "angle", "ans", "name", "component"], "variable_groups": [], "functions": {"otherway": {"parameters": [], "type": "html", "language": "javascript", "definition": "var div = Numbas.extensions.jsxgraph.makeBoard('500px','500px',{boundingBox:[-17,17,14,-14],grid:false,axis:false});\nvar board = div.board;\n\n//Doesn't look like you need this\nJXG.Options.text.useMathJax = true;\n\n// get the height of the triangle\nTT = Numbas.jme.unwrapValue(scope.variables.tritop);\ndx = Numbas.jme.unwrapValue(scope.variables.deltax);\nh = Numbas.jme.unwrapValue(scope.variables.h);\nv = Numbas.jme.unwrapValue(scope.variables.v);\nd = Numbas.jme.unwrapValue(scope.variables.d);\n\n\n// create the horizontal line\nvar hor = board.create('line',[[-12,-TT],[12,-TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the vertical line\nvar vert = board.create('line',[[-12,-TT],[-12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the diagonal line\nvar vert = board.create('line',[[12,-TT],[-12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the box for right angle\nboard.create('line',[[(-12+0.1*TT),-TT],[(-12+0.1*TT),-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\nboard.create('line',[[(-12+0.1*TT),-TT*0.9],[-12,-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//label the angle theta\nboard.create('text',[+6.2,-4*TT/5,\n  function() { \n    return '$\\\\theta$';\n  }], {fontSize:20,fixed: true});\n\n//label the angle phi\nboard.create('text',[-11,3*TT/5,\n  function() { \n    return '$\\\\phi$';\n  }], {fontSize:20,fixed: true});\n\n//display the side lengths\nvar vtext= board.create('text',[-15,0,v], {fontSize:20,fixed: true});\nvar htext= board.create('text',[-2,-TT-1,h], {fontSize:20,fixed: true});\nvar dtext= board.create('text',[-2,TT/2+0.3,d], {fontSize:20,fixed: true});\n\n//can't figure out how to rotate text. http://jsxgraph.uni-bayreuth.de/wiki/index.php/Texts_and_Transformations suggests the following\n//var tRot = board.create('transform', [Math.PI/2, 13,0], {type:'rotate'}); \n//tRot.bindTo(vtext);\n\n\n\n\nreturn div;\n\n"}, "triangle": {"parameters": [], "type": "html", "language": "javascript", "definition": "var div = Numbas.extensions.jsxgraph.makeBoard('500px','500px',{boundingBox:[-13,13,17,-17],grid:false,axis:false});\nvar board = div.board;\n\n//Doesn't look like you need this\nJXG.Options.text.useMathJax = true;\n\n// get the height of the triangle\nTT = Numbas.jme.unwrapValue(scope.variables.tritop);\ndx = Numbas.jme.unwrapValue(scope.variables.deltax);\nh = Numbas.jme.unwrapValue(scope.variables.h);\nv = Numbas.jme.unwrapValue(scope.variables.v);\nd = Numbas.jme.unwrapValue(scope.variables.d);\n\n\n// create the horizontal line\nvar hor = board.create('line',[[-12,-TT],[12,-TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the vertical line\nvar vert = board.create('line',[[12,-TT],[12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the diagonal line\nvar vert = board.create('line',[[-12,-TT],[12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the box for right angle\nboard.create('line',[[(12-0.1*TT),-TT],[(12-0.1*TT),-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\nboard.create('line',[[(12-0.1*TT),-TT*0.9],[12,-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//label the angle theta\nboard.create('text',[-7.2,-4*TT/5,\n  function() { \n    return '$\\\\theta$';\n  }], {fontSize:20,fixed: true});\n\n//label the angle phi\nboard.create('text',[9.6,3*TT/5,\n  function() { \n    return '$\\\\phi$';\n  }], {fontSize:20,fixed: true});\n\n//display the side lengths\nvar vtext= board.create('text',[12.5,0,v], {fontSize:20,fixed: true});\nvar htext= board.create('text',[0,-TT-1,h], {fontSize:20,fixed: true});\nvar dtext= board.create('text',[0,TT/2+0.3,d], {fontSize:20,fixed: true});\n\n//can't figure out how to rotate text. http://jsxgraph.uni-bayreuth.de/wiki/index.php/Texts_and_Transformations suggests the following\n//var tRot = board.create('transform', [Math.PI/2, 13,0], {type:'rotate'}); \n//tRot.bindTo(vtext);\n\n\n\n\nreturn div;\n\n"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{name[0]} wants to fire a rocket so that its {component[0][0]} velocity is {component[0][1]} $\\mathrm{ms^{-1}}$ and its {component[1][0]} velocity is {component[1][1]} $\\mathrm{ms^{-1}}$. At what angle from the horizontal should {name[1]} aim the rocket?</p>\n<p>[[0]] $^\\circ$ (to the nearest degree).</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": "50", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Vectors: Combinations defined or not 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": [], "tags": ["cross product", "dot product", "inner product", "scalar product", "scalars", "vector", "vector product", "vectors"], "preamble": {"css": "", "js": ""}, "advice": "\n    \n    \n    <p>1. $\\boldsymbol{(A\\cdot B)\\cdot C}$ is <b>undefined</b> as $\\boldsymbol{A\\cdot B}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{C}$.</p>\n    \n    \n    \n    <p>2. $\\boldsymbol{(A\\cdot B)C}$ is a <b>vector</b> and is a multiple of $\\boldsymbol{C}$ as $\\boldsymbol{A \\cdot B}$ is a scalar.</p>\n    \n    \n    \n    <p>3. $\\boldsymbol{(A\\cdot B)\\times C}$ is <b>undefined</b> as $\\boldsymbol{A\\cdot B}$ is a scalar and the cross product is only defined between vectors.</p>\n    \n    \n    \n    <p>4. $\\boldsymbol{(A\\times B)\\times C}$ is a <b>vector</b> as $\\boldsymbol{A \\times B}$ and $\\boldsymbol{C}$ are vectors and the cross product between vectors produces a vector.</p>\n    \n    \n    \n    <p>5. $\\boldsymbol{(A\\times B)\\cdot C}$ is a <b>scalar</b> as $\\boldsymbol{A \\times B}$ and $\\boldsymbol{C}$ are vectors and the inner or dot product is between vectors and produces a scalar.</p>\n    \n    \n  \n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"maxAnswers": 0, "shuffleChoices": true, "matrix": [[0, 0, 0.4], [0, 0.4, 0], [0, 0, 0.4], [0, 0.4, 0], [0.4, 0, 0]], "shuffleAnswers": true, "minAnswers": 0, "marks": 0, "answers": ["<p>Scalar</p>", "<p>Vector</p>", "<p>Undefined</p>"], "warningType": "none", "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "maxMarks": 0, "choices": ["<p>$\\boldsymbol{(A\\cdot B)\\cdot C}$</p>", "<p>$\\boldsymbol{(A\\cdot B)C}$</p>", "<p>$\\boldsymbol{(A\\cdot B)\\times C}$</p>", "<p>$\\boldsymbol{(A\\times B)\\times C}$</p>", "<p>$\\boldsymbol{(A\\times B)\\cdot C}$</p>"], "type": "m_n_x", "minMarks": 0}], "statement": "<p>Given the vectors $\\boldsymbol{A},\\;\\;\\boldsymbol{B}$ and $\\boldsymbol{C}$ in $3$ dimensional space, state whether the following quantities are scalars, vectors or undefined.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {}, "metadata": {"notes": "\n        \t\t    \t\t<p><strong>15/07/2012:</strong></p>\n        \t\t    \t\t<p>Added tags.</p>\n        \t\t    \t\t<p><strong>16/07/2012:</strong></p>\n        \t\t    \t\t<p><strong>Added tags.<br /></strong></p>\n        \t\t    \t\t<p><strong>&nbsp;</strong></p>\n        \t\t    \t\t<p><strong>&nbsp;</strong></p>\n        \t\t    \n        \t\t", "description": "<p>Determine if various combinations of vectors are defined or not.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Trigonometry and vectors: Find angles to nearest degree", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/undefined_YwBJcjH", "/srv/numbas/media/question-resources/undefined_YwBJcjH"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["epphys", "jsxgraph", "Jsxgraph", "JSXgraph", "Triangle", "triangle", "trig", "trigonometry", "vectors"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"anglelist": {"name": "anglelist", "group": "Ungrouped variables", "definition": "random(['$\\\\theta$',3],['$\\\\phi$',4])", "description": "", "templateType": "anything", "can_override": false}, "scale": {"name": "scale", "group": "Ungrouped variables", "definition": "24/triples[1]", "description": "", "templateType": "anything", "can_override": false}, "triples": {"name": "triples", "group": "Ungrouped variables", "definition": "random([[3, 4, 5], [5, 12, 13], [7, 24, 25], [8, 15, 17], [9, 40, 41], [11, 60, 61], [12, 35, 37], [16, 63, 65], [20, 21, 29], [20, 99, 101], [24, 143, 145], [28, 45, 53], [33, 56, 65], [36, 77, 85], [39, 80, 89], [44, 117, 125], [48, 55, 73], [51, 140, 149], [52, 165, 173], [57, 176, 185], [60, 91, 109], [60, 221, 229], [65, 72, 97], [68, 285, 293], [69, 260, 269], [75, 308, 317], [76, 357, 365], [84, 187, 205], [84, 437, 445], [85, 132, 157], [87, 416, 425], [88, 105, 137], [92, 525, 533], [93, 476, 485], [95, 168, 193], [96, 247, 265], [100, 621, 629], [104, 153, 185], [105, 208, 233], [105, 608, 617], [111, 680, 689], [115, 252, 277], [119, 120, 169], [120, 209, 241], [120, 391, 409], [132, 475, 493], [133, 156, 205], [135, 352, 377], [136, 273, 305], [140, 171, 221], [145, 408, 433], [152, 345, 377], [155, 468, 493], [156, 667, 685], [160, 231, 281], [161, 240, 289], [165, 532, 557], [168, 425, 457], [168, 775, 793], [175, 288, 337], [180, 299, 349], [184, 513, 545], [185, 672, 697], [189, 340, 389], [195, 748, 773], [200, 609, 641], [203, 396, 445], [204, 253, 325], [205, 828, 853], [207, 224, 305], [215, 912, 937], [216, 713, 745], [217, 456, 505], [220, 459, 509], [225, 272, 353], [228, 325, 397], [231, 520, 569], [232, 825, 857], [240, 551, 601], [248, 945, 977], [252, 275, 373], [259, 660, 709], [260, 651, 701], [261, 380, 461], [273, 736, 785], [276, 493, 565], [279, 440, 521], [280, 351, 449], [280, 759, 809], [287, 816, 865], [297, 304, 425], [300, 589, 661], [301, 900, 949], [308, 435, 533], [315, 572, 653], [319, 360, 481], [333, 644, 725], [336, 377, 505], [336, 527, 625], [341, 420, 541], [348, 805, 877], [364, 627, 725], [368, 465, 593], [369, 800, 881], [372, 925, 997], [385, 552, 673], [387, 884, 965], [396, 403, 565], [400, 561, 689], [407, 624, 745], [420, 851, 949], [429, 460, 629], [429, 700, 821], [432, 665, 793], [451, 780, 901], [455, 528, 697], [464, 777, 905], [468, 595, 757], [473, 864, 985], [481, 600, 769], [504, 703, 865], [533, 756, 925], [540, 629, 829], [555, 572, 797], [580, 741, 941], [615, 728, 953], [616, 663, 905], [696, 697, 985]])\n", "description": "<p>Some the following were too skinny and so were removed.<br /><br /><br />random([[3,4,5], [5,12,13], [7,24,25], [8,15,17], [9,40,41],<br />[11,60,61], [12,35,37], [13,84,85], [15,112,113], [16,63,65],<br /> [19,180,181], [20,21,29], [20,99,101],<br />[23,264,265], [24,143,145], [25,312,313], [27,364,365], [28,45,53],<br />[28,195,197], [31,480,481], [32,255,257], [33,56,65],<br />[33,544,545], [35,612,613], [36,77,85], [36,323,325], [37,684,685],<br />[39,80,89], [40,399,401], [41,840,841], [43,924,925],<br />[44,117,125], [44,483,485], [48,55,73], [48,575,577], [51,140,149],<br />[52,165,173], [52,675,677], [56,783,785], [57,176,185], [60,91,109],<br />[60,221,229], [60,899,901], [65,72,97], [68,285,293], [69,260,269],<br />[75,308,317], [76,357,365], [84,187,205], [84,437,445], [85,132,157],<br />[87,416,425], [88,105,137], [92,525,533], [93,476,485], [95,168,193],<br />[96,247,265], [100,621,629], [104,153,185], [105,208,233], [105,608,617],<br />[108,725,733], [111,680,689], [115,252,277], [116,837,845], [119,120,169],<br />[120,209,241], [120,391,409], [123,836,845], [129,920,929],<br />[132,475,493], [133,156,205], [135,352,377], [136,273,305], [140,171,221],<br />[145,408,433], [152,345,377], [155,468,493], [156,667,685], [160,231,281],<br />[161,240,289], [165,532,557], [168,425,457], [168,775,793], [175,288,337],<br />[180,299,349], [184,513,545], [185,672,697], [189,340,389], [195,748,773],<br />[200,609,641], [203,396,445], [204,253,325], [205,828,853], [207,224,305],<br />[215,912,937], [216,713,745], [217,456,505], [220,459,509], [225,272,353],<br />[228,325,397], [231,520,569], [232,825,857], [240,551,601], [248,945,977],<br />[252,275,373], [259,660,709], [260,651,701], [261,380,461], [273,736,785],<br />[276,493,565], [279,440,521], [280,351,449], [280,759,809], [287,816,865],<br />[297,304,425], [300,589,661], [301,900,949], [308,435,533], [315,572,653],<br />[319,360,481], [333,644,725], [336,377,505], [336,527,625], [341,420,541],<br />[348,805,877], [364,627,725], [368,465,593], [369,800,881], [372,925,997],<br />[385,552,673], [387,884,965], [396,403,565], [400,561,689], [407,624,745],<br />[420,851,949], [429,460,629], [429,700,821], [432,665,793], [451,780,901],<br />[455,528,697], [464,777,905], [468,595,757], [473,864,985], [481,600,769],<br />[504,703,865], [533,756,925], [540,629,829], [555,572,797], [580,741,941],<br />[615,728,953], [616,663,905], [696,697,985]])</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "if(vec1[2]=1,triples[2],'')", "description": "<p>use&nbsp;https://www.mathsisfun.com/numbers/pythagorean-triples.html</p>\n<p></p>\n<p>so always integer and we could scale by k for more randomness.</p>", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(vec1[0]=1,triples[0],'')", "description": "", "templateType": "anything", "can_override": false}, "deltax": {"name": "deltax", "group": "Ungrouped variables", "definition": "precround(-12+2*triples[1]/triples[0],4)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "if(vec1[1]=1,triples[1],'')", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "ansrad*180/(pi)", "description": "", "templateType": "anything", "can_override": false}, "component": {"name": "component", "group": "Ungrouped variables", "definition": "if(vec1=[1,1,0], [[\"vertical\",v], [\"horizontal\",h]],\nif(vec1=[1,0,1], [[\"vertical\",v], [\"total\",d]],\nif(vec1=[0,1,1], [[\"horizontal\",h],[\"total\",d]],'error')))", "description": "", "templateType": "anything", "can_override": false}, "name": {"name": "name", "group": "Ungrouped variables", "definition": "random([\"Ben\", \"he\"], [\"Annie\", \"she\"], [\"Matt\", \"he\"], [\"David\", \"he\"], [\"Steve\", \"he\"], [\"David\", \"he\"], [\"Scott\", \"he\"], [\"Fran\", \"she\"], [\"Jenny\", \"she\"], [\"Lyn\", \"she\"], [\"Judy-anne\", \"she\"], [\"Courtney\", \"she\"])", "description": "", "templateType": "anything", "can_override": false}, "ansrad": {"name": "ansrad", "group": "Ungrouped variables", "definition": "if(anglelist[1]=3,if(vec1[0]=0, arccos(h/d),if(vec1[1]=0,arcsin(v/d),arctan(v/h))),\nif(vec1[0]=0, arcsin(h/d),if(vec1[1]=0,arccos(v/d),arctan(h/v))))", "description": "", "templateType": "anything", "can_override": false}, "vec1": {"name": "vec1", "group": "Ungrouped variables", "definition": "shuffle([1,1,0])", "description": "", "templateType": "anything", "can_override": false}, "angle": {"name": "angle", "group": "Ungrouped variables", "definition": "anglelist[0]", "description": "", "templateType": "anything", "can_override": false}, "tritop": {"name": "tritop", "group": "Ungrouped variables", "definition": "precround(triples[0]*scale,4)/2", "description": "<p>top of triangle for jsxgraph, keeping same ratios.</p>", "templateType": "anything", "can_override": false}, "switcharoo": {"name": "switcharoo", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["triples", "vec1", "v", "h", "d", "scale", "tritop", "deltax", "switcharoo", "ansrad", "anglelist", "angle", "ans", "name", "component"], "variable_groups": [], "functions": {"otherway": {"parameters": [], "type": "html", "language": "javascript", "definition": "var div = Numbas.extensions.jsxgraph.makeBoard('500px','500px',{boundingBox:[-17,17,14,-14],grid:false,axis:false});\nvar board = div.board;\n\n//Doesn't look like you need this\nJXG.Options.text.useMathJax = true;\n\n// get the height of the triangle\nTT = Numbas.jme.unwrapValue(scope.variables.tritop);\ndx = Numbas.jme.unwrapValue(scope.variables.deltax);\nh = Numbas.jme.unwrapValue(scope.variables.h);\nv = Numbas.jme.unwrapValue(scope.variables.v);\nd = Numbas.jme.unwrapValue(scope.variables.d);\n\n\n// create the horizontal line\nvar hor = board.create('line',[[-12,-TT],[12,-TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the vertical line\nvar vert = board.create('line',[[-12,-TT],[-12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the diagonal line\nvar vert = board.create('line',[[12,-TT],[-12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the box for right angle\nboard.create('line',[[(-12+0.1*TT),-TT],[(-12+0.1*TT),-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\nboard.create('line',[[(-12+0.1*TT),-TT*0.9],[-12,-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//label the angle theta\nboard.create('text',[+6.2,-4*TT/5,\n  function() { \n    return '$\\\\theta$';\n  }], {fontSize:20,fixed: true});\n\n//label the angle phi\nboard.create('text',[-11,3*TT/5,\n  function() { \n    return '$\\\\phi$';\n  }], {fontSize:20,fixed: true});\n\n//display the side lengths\nvar vtext= board.create('text',[-15,0,v], {fontSize:20,fixed: true});\nvar htext= board.create('text',[-2,-TT-1,h], {fontSize:20,fixed: true});\nvar dtext= board.create('text',[-2,TT/2+0.3,d], {fontSize:20,fixed: true});\n\n//can't figure out how to rotate text. http://jsxgraph.uni-bayreuth.de/wiki/index.php/Texts_and_Transformations suggests the following\n//var tRot = board.create('transform', [Math.PI/2, 13,0], {type:'rotate'}); \n//tRot.bindTo(vtext);\n\n\n\n\nreturn div;\n\n"}, "triangle": {"parameters": [], "type": "html", "language": "javascript", "definition": "var div = Numbas.extensions.jsxgraph.makeBoard('500px','500px',{boundingBox:[-13,13,17,-17],grid:false,axis:false});\nvar board = div.board;\n\n//Doesn't look like you need this\nJXG.Options.text.useMathJax = true;\n\n// get the height of the triangle\nTT = Numbas.jme.unwrapValue(scope.variables.tritop);\ndx = Numbas.jme.unwrapValue(scope.variables.deltax);\nh = Numbas.jme.unwrapValue(scope.variables.h);\nv = Numbas.jme.unwrapValue(scope.variables.v);\nd = Numbas.jme.unwrapValue(scope.variables.d);\n\n\n// create the horizontal line\nvar hor = board.create('line',[[-12,-TT],[12,-TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the vertical line\nvar vert = board.create('line',[[12,-TT],[12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the diagonal line\nvar vert = board.create('line',[[-12,-TT],[12,TT]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//create the box for right angle\nboard.create('line',[[(12-0.1*TT),-TT],[(12-0.1*TT),-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\nboard.create('line',[[(12-0.1*TT),-TT*0.9],[12,-TT*0.9]], { straightFirst:false, straightLast:false, strokeColor: 'black', fixed: true});\n\n//label the angle theta\nboard.create('text',[-7.2,-4*TT/5,\n  function() { \n    return '$\\\\theta$';\n  }], {fontSize:20,fixed: true});\n\n//label the angle phi\nboard.create('text',[9.6,3*TT/5,\n  function() { \n    return '$\\\\phi$';\n  }], {fontSize:20,fixed: true});\n\n//display the side lengths\nvar vtext= board.create('text',[12.5,0,v], {fontSize:20,fixed: true});\nvar htext= board.create('text',[0,-TT-1,h], {fontSize:20,fixed: true});\nvar dtext= board.create('text',[0,TT/2+0.3,d], {fontSize:20,fixed: true});\n\n//can't figure out how to rotate text. http://jsxgraph.uni-bayreuth.de/wiki/index.php/Texts_and_Transformations suggests the following\n//var tRot = board.create('transform', [Math.PI/2, 13,0], {type:'rotate'}); \n//tRot.bindTo(vtext);\n\n\n\n\nreturn div;\n\n"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{name[0]} wants to fire a rocket so that its {component[0][0]} velocity is {component[0][1]} $\\mathrm{ms^{-1}}$ and its {component[1][0]} velocity is {component[1][1]} $\\mathrm{ms^{-1}}$. At what angle from the horizontal should {name[1]} aim the rocket?</p>\n<p>[[0]] $^\\circ$ (to the nearest degree).</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": 0, "precisionPartialCredit": "50", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Vectors: adding, multiply by scalar, magnitude", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h"], "tags": ["rebel", "Rebel", "REBEL", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>The vector $\\mathbf{x}+\\mathbf{y}$ is [[0]]$\\mathbf{i}+$[[1]]$\\mathbf{j}+$[[2]]$\\mathbf{k}$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{a}+{d}", "minValue": "{a}+{d}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 2, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{b}+{f}", "minValue": "{b}+{f}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 2, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{c}+{g}", "minValue": "{c}+{g}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 2, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>The vector {h}$\\mathbf{x}$ is [[0]]$\\mathbf{i}+$[[1]]$\\mathbf{j}+$[[2]]$\\mathbf{k}$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{h}*{a}", "minValue": "{h}*{a}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 2, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{h}*{b}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{h}*{c}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "precisionType": "dp", "prompt": "<p>Find $|\\mathbf{x}|$, the magnitude of $\\mathbf{x}$ to the nearest whole number.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "sqrt{a^2+b^2+c^2}", "strictPrecision": true, "minValue": "sqrt{a^2+b^2+c^2}", "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>If $\\mathbf{x}=x_1\\mathbf{i}+x_2\\mathbf{j}+x_3\\mathbf{k}$, then $|x|=\\sqrt{x_1^2+x_2^2+x_3^2}$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": 0, "scripts": {}, "marks": 8, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p>Let $\\mathbf{x}=${a}$\\mathbf{i}+${b}$\\mathbf{j}+${c}$\\mathbf{k}$ and $\\mathbf{y}=${d}$\\mathbf{i}+${f}$\\mathbf{j}+${g}$\\mathbf{k}$&nbsp;</p>\n<p></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(1..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "random(1..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(1..10)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}}, "metadata": {"description": "<p>Magnitude of a vector, adding vectors, multiply by a scalar.</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [["question-resources/undefined_YwBJcjH", "/srv/numbas/media/question-resources/undefined_YwBJcjH"]]}