// Numbas version: finer_feedback_settings
{"name": "Eryk's copy of MATH6005 Vectors", "duration": 0, "metadata": {"notes": "", "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"}, "allQuestions": true, "shuffleQuestions": false, "percentPass": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "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"}, "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": "Cartesian equation of a plane", "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/"}], "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "extendBaseMarkingAlgorithm": true, "prompt": "<p>We can write a&nbsp;vector equation of the plane in the form:</p>\n<p>$\\boldsymbol{r}=\\boldsymbol{r_1}+\\lambda (\\boldsymbol{r_2}-\\boldsymbol{r_1}) + \\mu (\\boldsymbol{r_3}-\\boldsymbol{r_1})$</p>\n<p>Note that three points determine a plane and</p>\n<ol>\n<li>$\\lambda=0,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_1}$</li>\n<li>$\\lambda=1,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_2}$</li>\n<li>$\\lambda=0,\\;\\;\\mu=1$ gives $\\boldsymbol{r}=\\boldsymbol{r_3}$</li>\n</ol>\n<p>Note that if we let</p>\n<p>\\[\\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1})\\]</p>\n<p>then $\\boldsymbol{n}\\cdot (\\boldsymbol{r_2}-\\boldsymbol{r_1})=0$ and $\\boldsymbol{n}\\cdot (\\boldsymbol{r_3}-\\boldsymbol{r_1})=0$.</p>\n<p>If $\\boldsymbol{r} = (x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that</p>\n<p>\\[ \\boldsymbol{r}\\cdot \\boldsymbol{n}=(x,\\;y,\\;z)\\cdot &nbsp;\\boldsymbol{n}=\\boldsymbol{r_1}\\cdot \\boldsymbol{n} \\]</p>", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true}], "prompt": "<p>Find the Cartesian equation of this plane, in the form $ax+by+cz = d$, with $a$, $b$ and $c$ integers, not decimals.</p>\n<p>Equation of the plane: [[0]]&nbsp;$ = $&nbsp;[[1]]</p>\n<p>You can get help by clicking on <em>Show steps</em>.</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "{coeffx}*x+{coeffy}*y + {coeffz}*z ", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "answerSimplification": "std", "expectedVariableNames": [], "showPreview": true, "notallowed": {"message": "<p>Input numbers as integers and not decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 2}, {"answer": "{con}", "showCorrectAnswer": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "expectedVariableNames": [], "showPreview": true, "checkVariableNames": false, "unitTests": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"coeffy": {"templateType": "anything", "group": "Answer", "definition": "n[1]", "name": "coeffy", "description": ""}, "b3": {"templateType": "anything", "group": "Vector r_2", "definition": "if(t=2,1,2)", "name": "b3", "description": ""}, "a": {"templateType": "anything", "group": "Vector r_1", "definition": "random(-9..9 except -1..1)", "name": "a", "description": ""}, "s6'": {"templateType": "anything", "group": "Vector r_2", "definition": "random(1,-1)", "name": "s6'", "description": ""}, "t": {"templateType": "anything", "group": "Vector r_1", "definition": "random(0..2)", "name": "t", "description": ""}, "directions": {"templateType": "anything", "group": "Vector r_2", "definition": "map(id(3)[x],x,shuffle(0..2))", "name": "directions", "description": ""}, "r_3": {"templateType": "anything", "group": "Vector r_3", "definition": "vector(random(-9..9 except -1..1),random(-9..9 except -1..1),random(2..9))", "name": "r_3", "description": ""}, "r_2": {"templateType": "anything", "group": "Vector r_2", "definition": "s6*directions[a3] + p1*directions[b3]", "name": "r_2", "description": ""}, "n": {"templateType": "anything", "group": "Answer", "definition": "cross(r_2-r_1,r_3-r_1)", "name": "n", "description": "<p>Normal to the plane</p>"}, "a3": {"templateType": "anything", "group": "Vector r_2", "definition": "if(t=0,1,0)", "name": "a3", "description": ""}, "r_1": {"templateType": "anything", "group": "Vector r_1", "definition": "id(3)[t]*random(-9..9 except -1..1)", "name": "r_1", "description": ""}, "coeffx": {"templateType": "anything", "group": "Answer", "definition": "n[0]", "name": "coeffx", "description": ""}, "coeffz": {"templateType": "anything", "group": "Answer", "definition": "n[2]", "name": "coeffz", "description": ""}, "s6": {"templateType": "anything", "group": "Vector r_2", "definition": "switch(\n  t=1,\n    if(r_3[2]*s6'=r_3[1]*p1,-s6',s6'),\n  t=2,\n    if(r_3[2]*s6'=r_3[0]*p1,-s6',s6'),\n    if(r_3[1]*s6'=r_3[0]*p1,-s6',s6')\n)", "name": "s6", "description": ""}, "p1": {"templateType": "anything", "group": "Vector r_2", "definition": "random(-5..5 except -1..1)", "name": "p1", "description": ""}, "con": {"templateType": "anything", "group": "Answer", "definition": "dot(r_1,n)", "name": "con", "description": ""}}, "ungrouped_variables": [], "functions": {}, "variable_groups": [{"variables": ["t", "a", "r_1"], "name": "Vector r_1"}, {"variables": ["a3", "b3", "s6'", "s6", "p1", "directions", "r_2"], "name": "Vector r_2"}, {"variables": ["r_3"], "name": "Vector r_3"}, {"variables": ["n", "coeffx", "coeffy", "coeffz", "con"], "name": "Answer"}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>A plane goes through the points:</p>\n<p>\\begin{align}<br/>\\boldsymbol{r_1} &amp;= \\var{r_1}, &amp; \\boldsymbol{r_2} &amp;= \\var{r_2}, &amp; \\boldsymbol{r_3} &amp;= \\var{r_3}<br/>\\end{align}</p>", "tags": ["cartesian equation of a plane", "checked2015", "equation of a plane", "plane through 3 points", "planes", "Steps", "steps", "the equation of a plane", "unused", "Vector", "vector", "vector equation of a plane", "vector geometry", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.</p>\n<p>There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.</p>"}, "advice": "<p>We can write a&nbsp;vector equation of the plane in the form:</p>\n<p>$\\boldsymbol{r}=\\boldsymbol{r_1}+\\lambda (\\boldsymbol{r_2}-\\boldsymbol{r_1}) + \\mu (\\boldsymbol{r_3}-\\boldsymbol{r_1})$</p>\n<p>Note that three points determine a plane and</p>\n<ol>\n<li>$\\lambda=0,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_1}$</li>\n<li>$\\lambda=1,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_2}$</li>\n<li>$\\lambda=0,\\;\\;\\mu=1$ gives $\\boldsymbol{r}=\\boldsymbol{r_3}$</li>\n</ol>\n<p>Note that if we let</p>\n<p>\\[\\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1})\\]</p>\n<p>then $\\boldsymbol{n}\\cdot (\\boldsymbol{r_2}-\\boldsymbol{r_1})=0$ and $\\boldsymbol{n}\\cdot (\\boldsymbol{r_3}-\\boldsymbol{r_1})=0$.</p>\n<p>Hence $\\boldsymbol{r}\\cdot&nbsp;\\boldsymbol{n}=\\boldsymbol{r_1}\\cdot&nbsp;\\boldsymbol{n}$.</p>\n<p>If $\\boldsymbol{r} = (x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that</p>\n<p>\\[ \\boldsymbol{r}\\cdot \\boldsymbol{n}=(x,\\;y,\\;z)\\cdot &nbsp;\\boldsymbol{n}=\\boldsymbol{r_1}\\cdot \\boldsymbol{n} \\]</p>\n<p>If $\\boldsymbol{r}=(x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that&nbsp;the equation of the plane is given by $\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}&nbsp;\\cdot \\boldsymbol{n} = \\boldsymbol{r_1} \\cdot \\boldsymbol{n}$.</p>\n<p>We have:</p>\n<p>\\[ \\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1}) = \\var{r_2-r_1} \\times \\var{r_3-r_1} = \\var{n} \\]</p>\n<p>Hence, $\\boldsymbol{r_1} \\cdot \\boldsymbol{n} = \\var{con}$.</p>\n<p>So the Cartesian equation of the plane is</p>\n<p>\\[ \\simplify[all,!noLeadingMinus]{{coeffx}x + {coeffy}y + {coeffz}z = {con}} \\]</p>"}, {"name": "Cartesian form of the equation of a plane", "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": ["x_0", "v", "w"], "name": "Vectors"}], "variables": {"w": {"group": "Vectors", "templateType": "anything", "definition": "vector(s3,p1,p3)", "name": "w", "description": ""}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "v": {"group": "Vectors", "templateType": "anything", "definition": "vector(0,s1,p2)", "name": "v", "description": ""}, "p1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s7*random(2..5)", "name": "p1", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s4", "description": ""}, "p3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(q3=p2*p1/s1,q3+1,q3)", "name": "p3", "description": ""}, "x_0": {"group": "Vectors", "templateType": "anything", "definition": "vector(pm5,0,0)", "name": "x_0", "description": ""}, "s7": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s7", "description": ""}, "q3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..5)", "name": "q3", "description": ""}, "p2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s7*random(2..5)", "name": "p2", "description": ""}, "pm5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s6*random(2..5)", "name": "pm5", "description": ""}, "coeffx": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*p3-p2*p1", "name": "coeffx", "description": ""}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s3", "description": ""}, "s6": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s6", "description": ""}, "coeffz": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-s1*s3", "name": "coeffz", "description": ""}, "con": {"group": "Ungrouped variables", "templateType": "anything", "definition": "pm5*(s1*p3-p2*p1)", "name": "con", "description": ""}, "coeffy": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p2*s3", "name": "coeffy", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}}, "ungrouped_variables": ["p2", "pm5", "q3", "p1", "p3", "s3", "s2", "s1", "s7", "s6", "s4", "coeffz", "coeffy", "coeffx", "con"], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showFeedbackIcon": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>The vector equation of a plane is</p>\n<p>\\[ \\boldsymbol{x}=\\boldsymbol{x_0}+\\lambda \\boldsymbol{v} + \\mu \\boldsymbol{w} \\]</p>\n<p>If you let&nbsp;$\\boldsymbol{n}=\\boldsymbol{v} \\times \\boldsymbol{w}$, then $\\boldsymbol{x} \\cdot \\boldsymbol{n} = \\boldsymbol{x_0} \\cdot \\boldsymbol{n}$ as $\\boldsymbol{v}\\cdot \\boldsymbol{n}=0$ and $\\boldsymbol{w}\\cdot \\boldsymbol{n}=0$.</p>\n<p>If $\\boldsymbol{x}=(x,\\;y,\\;z)$ is the Cartesian representation of a point $\\boldsymbol{x}$ on the plane, the equation of the plane in Cartesian coordinates is then given by:</p>\n<p>\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}&nbsp;\\cdot \\boldsymbol{n} =\\boldsymbol{x_0} \\cdot \\boldsymbol{n}\\]</p>", "marks": 0, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "unitTests": []}], "prompt": "<p>Enter $ax + by + cz$ with $a$, $b$ and $c$ integers, not decimals.</p>\n<p>Equation of the plane: [[0]]&nbsp;$ = $&nbsp;[[1]]</p>\n<p>You can get help by clicking on <em>Show steps</em>.</p>", "stepsPenalty": 0, "unitTests": [], "scripts": {}, "gaps": [{"answer": "{coeffx}*x+{coeffy}*y + {coeffz}*z ", "showCorrectAnswer": true, "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "std", "showPreview": true, "checkVariableNames": false, "unitTests": [], "notallowed": {"message": "<p>Input numbers as integers and not decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "vsetRange": [0, 1], "vsetRangePoints": 5, "checkingType": "absdiff", "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "expectedVariableNames": [], "variableReplacements": [], "marks": "1.5", "showFeedbackIcon": true}, {"answer": "{con}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "showPreview": true, "failureRate": 1, "checkVariableNames": false, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "checkingType": "absdiff", "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.5", "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>You are given the vector equation of a plane in $\\mathbb{R}^3$:</p>\n<p>\\[ \\boldsymbol{x} = \\var{x_0} + \\lambda \\var{v} + \\mu \\var{w}, \\quad -\\infty\\lt\\lambda,\\;\\mu \\lt \\infty \\]</p>\n<p>In this question you want to find an equation of this plane in the Cartesian from</p>\n<p>\\[ ax + by + cz = d&nbsp;\\]</p>", "tags": ["Cartesian form of the equation of a plane", "checked2015", "cross product", "equation of a plane", "Parametric form for a plane in three space", "unused", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the Cartesian form $ax+by+cz=d$ of the equation of the plane $\\boldsymbol{r=r_0+\\lambda a+\\mu b}$.</p>\n<p>The solution is not unique. The constant on right hand side could be given to ensure that the left hand side is unique.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The vector equation of the plane is</p>\n<p>\\[ \\boldsymbol{x}=\\boldsymbol{x_0}+\\lambda \\boldsymbol{v} + \\mu \\boldsymbol{w} \\]</p>\n<p>where</p>\n<p>\\begin{align}<br/>\\boldsymbol{x_0} &amp;= \\var{x_0}, &amp; \\boldsymbol{v} &amp;= \\var{v}, &amp; \\boldsymbol{w} &amp;= \\var{w}<br/>\\end{align}</p>\n<p>We have</p>\n<p>\\[ \\boldsymbol{n} = \\boldsymbol{v} \\times \\boldsymbol{w} = \\var{v} \\times \\var{w} = \\var{cross(v,w)} \\]</p>\n<p>If you let&nbsp;$\\boldsymbol{n}=\\boldsymbol{v} \\times \\boldsymbol{w}$, then $\\boldsymbol{x} \\cdot \\boldsymbol{n} = \\boldsymbol{x_0} \\cdot \\boldsymbol{n}$ as $\\boldsymbol{v}\\cdot \\boldsymbol{n}=0$ and $\\boldsymbol{w}\\cdot \\boldsymbol{n}=0$.</p>\n<p>If $\\boldsymbol{x}=(x,\\;y,\\;z)$ is the Cartesian representation of a point $\\boldsymbol{x}$ on the plane, the equation of the plane in Cartesian coordinates is then given by:</p>\n<p>\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}&nbsp;\\cdot \\boldsymbol{n} =\\boldsymbol{x_0} \\cdot \\boldsymbol{n}\\]</p>\n<p>That is,</p>\n<p>\\[ \\simplify[all,!noLeadingMinus]{ {coeffx}*x+{coeffy}*y + {coeffz}*z } = \\var{con} \\]</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": "Find the angle between planes", "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/"}], "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$\\alpha = $ [[0]]&nbsp;radians</p>\n<p><em>(Enter your answer in radians, to 3 decimal places)</em></p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "mustBeReduced": false, "type": "numberentry", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "allowFractions": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "minValue": "ans", "maxValue": "ans", "precision": 3, "unitTests": [], "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "marks": 2, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.001", "description": "", "name": "tol"}, "r1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(-1,1)*random(1..9),3))", "description": "", "name": "r1"}, "n1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(r2-r1,r3-r1)", "description": "", "name": "n1"}, "tn1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(r2-r1,tr3-r1)", "description": "", "name": "tn1"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "b1"}, "tr3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(sign(r1[0])*random(1..9),sign(r1[1])*random(1..5),sign(r1[2])*random(1..9))", "description": "", "name": "tr3"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "a1"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "d1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "c1"}, "r3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if (tn1=vector(0), vector(tr3[0]+1,tr3[1],tr3[2]), tr3)", "description": "", "name": "r3"}, "ans": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(alpha>pi/2,pi-alpha,alpha)", "description": "", "name": "ans"}, "r2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(sign(r1[0])*random(1..9 except abs(r1[0])),sign(r1[1])*random(2..5),sign(r1[2])*random(1..5))", "description": "", "name": "r2"}, "alpha": {"group": "Ungrouped variables", "templateType": "anything", "definition": "arccos((a1*n1[0]+b1*n1[1]+c1*n1[2])/sqrt((abs(n1)^2)*(a1^2+b1^2+c1^2)))", "description": "", "name": "alpha"}}, "ungrouped_variables": ["tn1", "r1", "r2", "r3", "a1", "b1", "tol", "alpha", "ans", "n1", "c1", "tr3", "d1"], "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Calculate the angle $\\alpha,\\;0\\leq\\alpha \\leq \\frac{\\pi}{2}$, between the plane $\\Pi_1$, passing through the points</p>\n<p>\\begin{align}<br/>\\boldsymbol{r_1} &amp;= \\var{r1}, &amp; \\boldsymbol{r_2} &amp;= \\var{r2}, &amp;&nbsp;\\boldsymbol{r_3} &amp;= \\var{r3}<br/>\\end{align}</p>\n<p>and the plane, $\\Pi_2$, whose equation is</p>\n<p>\\[\\simplify[std]{{a1}x+{b1}y+{c1}z={d1}}\\]</p>", "tags": ["angle between lines with a common point in 3 space", "angle between planes", "cartesian equation of a plane", "checked2015", "diagram needed", "finding the angle between two planes", "normal to a plane", "parametric form of a plane", "plane given by three points in three space", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find angle between plane $\\Pi_1$, given by three points, and the plane $\\Pi_2$ given in Cartesian form.</p>\n<p>The calculation of $cos(\\alpha)$ at the end of Advice has fractionNumbers switched on and so the result is presented as a fraction, which can be misleading. Best if calculation is followed through without using fractionNumbers.</p>"}, "advice": "<p>The angle between two planes is given by the angle between their normals.</p>\n<p>The plane $\\Pi_1$ can be written in the form</p>\n<p>\\[\\boldsymbol{r} = \\boldsymbol{r}_1+\\lambda( \\boldsymbol{r}_2 - \\boldsymbol{r}_1)+\\mu( \\boldsymbol{r}_3- \\boldsymbol{r}_1)\\]</p>\n<p>and the normal $\\boldsymbol{n}_1$ to this plane is given by:</p>\n<p>\\[ \\boldsymbol{n}_1 = (\\boldsymbol{r}_2 - \\boldsymbol{r}_1)\\times (\\boldsymbol{r}_3- \\boldsymbol{r}_1)\\]</p>\n<p>For $ \\boldsymbol{r}_1$, $\\boldsymbol{r}_2$, $\\boldsymbol{r}_3$ as given.</p>\n<p>\\[ \\boldsymbol{n}_1 = \\var{n1}&nbsp;\\]</p>\n<p>The normal to the plane $\\Pi_2$ is given by</p>\n<p>\\[ \\boldsymbol{n}_2 = \\var{vector(a1,b1,c1)} \\]</p>\n<p>The angle between the two normals (and hence the two planes) can be found using:</p>\n<p>\\[ \\cos(\\alpha) = \\frac{\\boldsymbol{n}_1 \\cdot \\boldsymbol{n}_2}{\\lVert\\boldsymbol{n}_1\\rVert \\lVert\\boldsymbol{n}_2\\rVert} \\]</p>\n<p>On calculating this, we&nbsp;obtain</p>\n<p>\\begin{align}<br/>\\boldsymbol{n}_1 \\boldsymbol{\\cdot} \\boldsymbol{n}_2 &amp;= \\var{a1*n1[0]+b1*n1[1]+c1*n1[2]} \\\\<br/>\\lVert\\boldsymbol{n}_1\\rVert &amp;= \\simplify[std]{sqrt({n1[0]^2+n1[1]^2+n1[2]^2})} \\\\<br/> \\lVert\\boldsymbol{n}_2\\rVert &amp;= \\simplify[std]{sqrt({a1^2+b1^2+c1^2})}\\\\<br/>\\cos(\\alpha) &amp;= \\simplify[std]{{(a1*n1[0]+b1*n1[1]+c1*n1[2])/sqrt((n1[0]^2+n1[1]^2+n1[2]^2)*(a1^2+b1^2+c1^2))}}<br/>\\end{align}</p>\n<p>Now calculate $\\arccos(\\alpha) = \\var{precround(alpha,3)}$. The angle returned by your calculator will give a value between $0$ and $\\pi$. If it's bigger than $\\frac{\\pi}{2}$, subtract the&nbsp;calculated value from $\\pi$ to obtain an acute angle. So the angle between the two planes is&nbsp;</p>\n<p><span data-jme-visible=\"alpha&gt;pi/2\">\\[ \\alpha' = \\pi - \\var{precround(alpha,3)} = \\var{precround(ans,3)} \\text{ radians} \\]</span></p>\n<p><span data-jme-visible=\"alpha&lt;=pi/2\">\\[ \\alpha = \\var{precround(ans,3)} \\text{ radians} \\]</span></p>\n<p>to 3 decimal places.</p>\n<p></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": "Minimum distance between a point and a line in 3D", "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": ["directions", "p"], "name": "Vector p"}, {"variables": ["s1", "a", "b", "g", "x_0"], "name": "Vector x_0"}, {"variables": ["c", "d", "f", "v"], "name": "Vector v"}, {"variables": ["top"], "name": "Answer"}], "variables": {"s1": {"group": "Vector x_0", "templateType": "anything", "definition": "random(-1,1)", "description": "", "name": "s1"}, "v": {"group": "Vector v", "templateType": "anything", "definition": "vector(random(-9..9 except -1..1),random(-9..9 except -1..1),random(2..9))", "description": "", "name": "v"}, "d": {"group": "Vector v", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "d"}, "top": {"group": "Answer", "templateType": "anything", "definition": "cross(p-x_0,v)", "description": "<p>Top of the formula for the minimum distance: $(p-x_0) \\times v$.</p>", "name": "top"}, "f": {"group": "Vector v", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "f"}, "directions": {"group": "Vector p", "templateType": "anything", "definition": "map(id(3)[x],x,shuffle(0..2))", "description": "<p>Shuffled list of axis&nbsp;vectors</p>", "name": "directions"}, "b": {"group": "Vector x_0", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "b"}, "a": {"group": "Vector x_0", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "g": {"group": "Vector x_0", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}, "p": {"group": "Vector p", "templateType": "anything", "definition": "random(-1,1)*directions[0] + random(-5..5 except -1..1)*directions[1]", "description": "", "name": "p"}, "x_0": {"group": "Vector x_0", "templateType": "anything", "definition": "vector(random(2..9),0,random(2..9))*random(-1,1) + vector(0,random(2..9),0)*random(-1,1)", "description": "", "name": "x_0"}, "c": {"group": "Vector v", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "c"}}, "ungrouped_variables": [], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "showCorrectAnswer": true, "unitTests": [], "prompt": "<p>The minimum distance between the line and the point is given by</p>\n<p>\\[ \\frac{\\left\\lVert(\\boldsymbol{p} - \\boldsymbol{x_0})\\times \\boldsymbol{v} \\right\\rVert}{\\lVert \\boldsymbol{v} \\rVert}\\]</p>", "variableReplacements": [], "marks": 0}], "prompt": "<p>Distance = [[0]]</p>\n<p>Enter your answer exactly, using the function <code>sqrt(x)</code> if necessary. Do not use decimals.</p>\n<p>You can get help by clicking on <em>Show steps</em>.</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "sqrt({abs(top)^2}/{abs(v)^2})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "showPreview": true, "notallowed": {"message": "<p>Enter all numbers as integers, do not use decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "vsetRangePoints": 5, "failureRate": 1, "scripts": {}, "answerSimplification": "std", "type": "jme", "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>In $\\mathbb{R}^3$ find the distance between the point $\\boldsymbol{p} = \\var{p}$ and the line through the point $\\boldsymbol{x_0} = \\var{x_0}$ that is parallel to the vector $\\boldsymbol{v} = \\var{v}$.</p>", "tags": ["checked2015", "cross product of vectors", "distance between a point and a line", "distance between two points", "equation of a line through a point and in the direction of a vector", "minimum distance", "minimum distance between a point and a line in three space", "modulus of a vector", "three dimensional vector geometry", "vector", "Vector", "vector geometry", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find minimum distance between a point and a line in 3-space. The line goes through a given point in the direction of a given vector.</p>\n<p>The correct solution is given, however the accuracy of 0.001&nbsp; is not enough as in some cases answers near to the correct solution are also marked as correct.</p>"}, "advice": "<p>The line through $\\boldsymbol{x_0} = \\var{x_0}$ in the direction of $\\boldsymbol{v}=\\var{v}$ has equation:</p>\n<p>\\[ \\boldsymbol{r} = \\boldsymbol{x_0} + \\lambda \\boldsymbol{v} = \\var{x_0} + \\lambda \\var{v} \\]</p>\n<p>The minimum distance between this line and the point $\\boldsymbol{p} = \\var{p}$ is given by</p>\n<p>\\[ \\frac{\\left\\lVert(\\boldsymbol{p}-\\boldsymbol{x_0}) \\times \\boldsymbol{v} \\right\\rVert}{\\lVert \\boldsymbol{v} \\rVert} \\]</p>\n<p>Now,</p>\n<p>\\begin{align}<br/>\\boldsymbol{p} - \\boldsymbol{x_0} &amp;= \\var{p-x_0}&nbsp;\\Rightarrow \\\\[1em]<br/>(\\boldsymbol{p}-\\boldsymbol{x_0}) \\times \\boldsymbol{v} &amp;= \\var{p-x_0} \\times \\var{v} \\\\[1em]<br/>&amp;= \\var{top}<br/>\\end{align}</p>\n<p>Since</p>\n<p>\\begin{align}<br/>\\left\\lVert \\begin{matrix} \\var{top[0]} \\\\ \\var{top[1]} \\\\ \\var{top[2]} \\end{matrix}&nbsp;\\right \\rVert&nbsp;&amp;= \\simplify[]{sqrt({top[0]}^2 + {top[1]}^2 + {top[2]}^2)} \\\\ &amp;= \\sqrt{\\var{abs(top)^2}}<br/>\\end{align}</p>\n<p>and $\\lVert \\boldsymbol{v} \\rVert = \\simplify[]{sqrt({v[0]}^2 + {v[1]}^2 + {v[2]}^2)} = \\sqrt{\\var{abs(v)^2}}$,&nbsp;the distance is then:</p>\n<p>\\[\\simplify{sqrt({abs(top)^2}/{abs(v)^2})}\\]</p>"}, {"name": "Scalar triple product of coplanar 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/"}], "variable_groups": [{"variables": ["w1", "w2", "w3"], "name": "w"}, {"variables": ["s1", "s2", "s3", "s4", "s5", "s6"], "name": "s"}, {"variables": ["x1", "x2", "x3"], "name": "x"}, {"variables": ["y1", "y2", "y3"], "name": "y"}, {"variables": ["z1", "z2"], "name": "z"}], "variables": {"s1": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "x2": {"group": "x", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "x2"}, "y3": {"group": "y", "templateType": "anything", "definition": "if(x1=x3,if(y1=ty3,-ty3,ty3),ty3)", "description": "", "name": "y3"}, "w1": {"group": "w", "templateType": "anything", "definition": "x2*y3-y2*x3", "description": "", "name": "w1"}, "s4": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "y2": {"group": "y", "templateType": "anything", "definition": "if(x1*z2=x2*y1,z2+1,z2)", "description": "", "name": "y2"}, "s2": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "z2": {"group": "z", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "z2"}, "x3": {"group": "x", "templateType": "anything", "definition": "random(-9..9)", "description": "", "name": "x3"}, "lambda": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(x3*(y1*z2-y2*z1)+y3*(x2*z1-x1*z2))/(x2*y1-x1*y2)", "description": "", "name": "lambda"}, "w2": {"group": "w", "templateType": "anything", "definition": "x3*y1-y3*x1", "description": "", "name": "w2"}, "x1": {"group": "x", "templateType": "anything", "definition": "s1*random(1..9)", "description": "", "name": "x1"}, "s5": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s5"}, "s3": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "s6": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s6"}, "y1": {"group": "y", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "y1"}, "z1": {"group": "z", "templateType": "anything", "definition": "s5*random(1..9)", "description": "", "name": "z1"}, "w3": {"group": "w", "templateType": "anything", "definition": "x1*y2-x2*y1", "description": "", "name": "w3"}, "ty3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s6*random(1..9)", "description": "", "name": "ty3"}}, "ungrouped_variables": ["ty3", "lambda"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "allowFractions": true, "type": "numberentry", "showCorrectAnswer": true, "minValue": "lambda", "correctAnswerFraction": true, "variableReplacements": [], "marks": 2, "maxValue": "lambda"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\lambda=$&nbsp;[[0]].</p>\n<p>Enter&nbsp;your answer as a&nbsp;fraction or integer and not a decimal.</p>", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are given three points in $\\mathbb{R}^3$,</p>\n<p>\\begin{align}<br/>\\boldsymbol{a} &amp;= \\var{vector(x1,x2,x3)}, &amp;<br/>\\boldsymbol{b} &amp;= \\var{vector(y1,y2,y3)}, &amp;<br/>\\boldsymbol{c} &amp;= \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix}<br/>\\end{align}</p>\n<p>where $\\lambda$ is a parameter to be determined.</p>\n<p>Find the&nbsp;value of $\\lambda$ such that $\\boldsymbol{a}$, $\\boldsymbol{b}$ and $\\boldsymbol{c}$ all lie on the same plane through the origin.</p>", "tags": ["checked2015", "colinear", "colinearvectors", "cross product", "determining if three vectors in three space are colinear", "dot product", "inner product", "scalar product", "scalar triple product", "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 module tag</p>\n<p><strong></strong></p>\n<p><strong>03/12/2013</strong></p>\n<p>Clarified wording vector -&gt; position vector. (AJY)</p>\n<p>Typo colinear -&gt; collinear. (AJY)</p>\n<p><strong>15/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Corrected error. It was possible for the third coordinate of $A \\times B$ &nbsp;to be 0. Hence could not be colinear in some circumstances.&nbsp;Calculations checked as a result.</p>\n<p><strong>16/07/2012:</strong></p>\n<p>Added tags.<strong><br /></strong></p>\n<p>Moved \\rightarrow so that it is located at the beginning of the line.</p>\n<p>Question appears to be working correctly.</p>\n<p><strong>27/08/2012:</strong></p>\n<p>Changed coplanar to colinear - for obvious reasons!!</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Three 3 dim vectors, one with a parameter $\\lambda$ in the third coordinate. Find value of $\\lambda$ ensuring vectors coplanar. Scalar triple product.</p>"}, "advice": "<p>Note that $\\boldsymbol{a}\\times \\boldsymbol{b}$ is a vector which is perpendicular to both $\\boldsymbol{a}$ and $\\boldsymbol{b}$ and hence to the plane through the origin&nbsp;containing $\\boldsymbol{a}$ and $\\boldsymbol{b}$.&nbsp;</p>\n<p>So if $\\boldsymbol{c}$ is perpendicular to $\\boldsymbol{a} \\times \\boldsymbol{b}$, i.e. $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} = 0$, it must lie on the same plane.</p>\n<p>Now</p>\n<p>\\begin{align}&nbsp;<br/>\\boldsymbol{a} \\times \\boldsymbol{b} &amp;= \\var{vector(x1,x2,x3)} \\times \\var{vector(y1,y2,y3)} \\\\[1em]<br/> &amp;= \\simplify[]{vector({x2}*{y3}-{x3}*{y2}, {x3}*{y1}-{x1}*{y3}, {x1}*{y2}-{x2}*{y1})} \\\\[1em]<br/>&amp;= \\var{vector(w1,w2,w3)}<br/>\\end{align}</p>\n<p>Hence</p>\n<p>\\begin{align}<br/>(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &amp;= \\var{vector(w1,w2,w3)} \\boldsymbol{\\cdot} \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix} \\\\[1em]<br/>&amp;= \\simplify[]{{w1}*{z1}+{w2}*{z2}+{w3}*lambda} \\\\[1em]<br/>&amp;= \\simplify{{w1*z1+w2*z2}+{w3}*lambda}<br/>\\end{align}</p>\n<p><br/>We now require a value of $\\lambda$ so that $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c}=0$.</p>\n<p>\\begin{align}<br/>&amp;&amp;(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &amp;= &nbsp;0 \\\\<br/>\\implies &amp;&amp;\\simplify{{w1*z1+w2*z2}+{w3}*lambda} &amp;= 0 \\\\<br/> \\implies &amp;&amp;\\lambda &amp;= \\simplify[std]{{-w1*z1-w2*z2}/{w3}}<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": "Vector equation of a line, ", "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/"}], "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Find the vector equation of Line 1 which goes through the point $\\boldsymbol{x_0}$ in the direction of the vector $\\boldsymbol{v}$.</p>\n<p>Input the vector equation in the form:</p>\n<p>\\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix}&nbsp;+ \\lambda \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} \\]</p>\n<p>such that $\\boldsymbol{r} = \\boldsymbol{x_0}$ when $\\lambda=0$ and $\\boldsymbol{r}=\\boldsymbol{x_0}+\\boldsymbol{v}$ when $\\lambda=1$ by filling in the appropriate fields below:</p>\n<p>$ \\boldsymbol{r} = $&nbsp;[[0]]&nbsp;$ + \\lambda $&nbsp;[[1]]</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(a,b,g)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(c,d,f)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Once again find the vector equation of Line 2 which goes through the point $\\boldsymbol{y_0}$ in the direction of the vector $\\boldsymbol{w}$ in the form</p>\n<p>\\[ \\boldsymbol{r} = \\begin{pmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{pmatrix} + \\mu \\begin{pmatrix} d_1 \\\\ d_2 \\\\ d_3 \\end{pmatrix} \\]</p>\n<p>such that $\\boldsymbol{r}=\\boldsymbol{C}$ when $\\mu=0$ and $\\boldsymbol{r=C+D}$ when $\\mu=1$ by filling in the appropriate fields below:</p>\n<p>$ \\boldsymbol{r} = $&nbsp;[[0]] $ + \\mu $&nbsp;[[1]]</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(a1,b1,g1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(c1,d1,f1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "<p>You are told&nbsp;that Line 1 and Line 2 intersect in a point $\\boldsymbol{P}$.</p>\n<p>Find $\\boldsymbol{P}$.</p>\n<p>$\\boldsymbol{P} = $&nbsp;[[0]]</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "p", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "3", "numRows": "3"}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([c,d,f])", "name": "w", "description": ""}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([a,b,g])", "name": "v", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "name": "b", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "ga", "name": "f1", "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": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+lam*c-mu*al", "name": "a1", "description": ""}, "mu": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..5)", "name": "mu", "description": ""}, "be": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "name": "be", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "a", "description": ""}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(a,b,g)+lam*vector(c,d,f)", "name": "p", "description": "<p>Point of intersection of the two lines</p>"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "name": "c", "description": ""}, "g1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "g+lam*f-mu*ga", "name": "g1", "description": ""}, "lam": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..5)", "name": "lam", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "name": "d", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "f", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b+lam*d-mu*be", "name": "b1", "description": ""}, "al": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "name": "al", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "be", "name": "d1", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "al", "name": "c1", "description": ""}, "ga": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "name": "ga", "description": ""}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s3", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "g", "description": ""}}, "ungrouped_variables": ["a", "a1", "al", "b", "b1", "be", "c", "c1", "d", "d1", "f", "f1", "g", "g1", "ga", "lam", "mu", "s1", "s2", "s3", "s4", "v", "w", "p"], "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are given the vectors</p>\n<p>\\begin{align}<br/>\\boldsymbol{x_0} &amp;= \\var{vector(a,b,g)} , &amp; \\boldsymbol{v} &amp; = \\var{vector(c,d,f)}, \\\\[1em]<br/>\\boldsymbol{y_0} &amp;= \\var{vector(a1,b1,g1)}, &amp; \\boldsymbol{w} &amp;=\\var{vector(c1,d1,f1)}<br/>\\end{align}</p>\n<p>in $\\mathbb{R^3}$.</p>", "tags": ["checked2015", "equation of a line", "equation of a line through a vector in the direction of another vector", "Finding a common point for two lines in three dimensional space", "intersection of two lines in three dimensional space", "lines in three dimensional space", "three dimensional geometry", "vector equation of a line", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given two 3 dim vectors,&nbsp;find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.</p>"}, "advice": "<h4>a)</h4>\n<p>\\[\\boldsymbol{r} = \\var{vector(a,b,g)} + \\lambda \\var{vector(c,d,f)}\\]</p>\n<h4>b)</h4>\n<p>\\[\\boldsymbol{r} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\\]</p>\n<h4>c)</h4>\n<p>Write out a set of simultaneous equations for each component of $\\boldsymbol{P}$:</p>\n<p>\\begin{align}<br/>\\simplify[]{{a} + lambda*{c}} &amp;= \\simplify[]{{a1} + mu*{c1}} \\\\<br/>\\simplify[]{{b} + lambda*{d}} &amp;= \\simplify[]{{b1} + mu*{d1}} \\\\<br/>\\simplify[]{{g} + lambda*{f}} &amp;= \\simplify[]{{g1} + mu*{f1}}<br/>\\end{align}</p>\n<p>By solving these equations, we find that the point $\\boldsymbol{P}$ common to both lines is given by $\\lambda=\\var{lam},\\mu=\\var{mu}$, and</p>\n<p>\\[\\boldsymbol{P} = \\var{p}\\]</p>"}]}], "contributors": [{"name": "Eryk Wilczy?ski", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/781/"}], "extensions": [], "custom_part_types": [], "resources": []}