// Numbas version: exam_results_page_options
{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Questions on vector arithmetic and vector operations, including dot and cross product.</p>"}, "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-ordered", "questions": [{"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": []}]}, {"name": "Elementary operations on vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katy Dobson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/854/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Paul Emanuel", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2551/"}], "ungrouped_variables": ["b4", "q", "s3", "s2", "s1", "s5", "s4", "ssquares", "v1", "v2", "a4", "a", "c", "b", "d", "g", "f", "m", "n", "ssquaresb", "ssquaresa", "v"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "tags": [], "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "gaps": [{"type": "matrix", "allowResize": false, "allowFractions": false, "correctAnswerFractions": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "numColumns": 1, "variableReplacements": [], "customMarkingAlgorithm": "", "correctAnswer": "v", "tolerance": 0, "markPerCell": false, "numRows": "3", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Calculate&nbsp;$\\mathbf{v}+\\mathbf{w} = $ [[0]]</p>", "scripts": {}}, {"type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "gaps": [{"type": "matrix", "allowResize": false, "allowFractions": false, "correctAnswerFractions": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "numColumns": 1, "variableReplacements": [], "customMarkingAlgorithm": "", "correctAnswer": "v1 + 3*v2", "tolerance": 0, "markPerCell": false, "numRows": "3", "extendBaseMarkingAlgorithm": true, "marks": 1, "showFeedbackIcon": true, "scripts": {}}], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Calculate&nbsp;$\\mathbf{v}+3\\mathbf{w} = $[[0]]</p>", "scripts": {}}, {"type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "gaps": [{"type": "matrix", "allowResize": false, "allowFractions": false, "correctAnswerFractions": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "numColumns": 1, "variableReplacements": [], "customMarkingAlgorithm": "", "correctAnswer": "4*v1 - 2*v2", "tolerance": 0, "markPerCell": false, "numRows": "3", "extendBaseMarkingAlgorithm": true, "marks": 1, "showFeedbackIcon": true, "scripts": {}}], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Calculate&nbsp;$4\\mathbf{v}-2\\mathbf{w} = $[[0]]</p>", "scripts": {}}, {"type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "gaps": [{"checkVariableNames": false, "type": "jme", "vsetRange": [0, 1], "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "answer": "sqrt({a^2+b^2+g^2})", "checkingType": "absdiff", "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "answerSimplification": "std", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}, {"checkVariableNames": false, "type": "jme", "vsetRange": [0, 1], "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "answer": "sqrt({c^2+d^2+f^2})", "checkingType": "absdiff", "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "answerSimplification": "std", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}, {"checkVariableNames": false, "type": "jme", "vsetRange": [0, 1], "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "answer": "sqrt({(a+c)^2+ (b+d)^2 +(g+f)^2})", "checkingType": "absdiff", "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "answerSimplification": "std", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Calculate the following.</p>\n<p>$\\vert \\mathbf{v} \\vert=$&nbsp;[[0]]</p>\n<p>$\\vert \\mathbf{w} \\vert = $&nbsp;[[1]]</p>\n<p>$\\vert \\mathbf{v}+\\mathbf{w} \\vert = $&nbsp;[[2]]</p>", "scripts": {}}, {"type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "gaps": [{"checkVariableNames": false, "type": "jme", "vsetRange": [0, 1], "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "answer": "({(a + c)} / Sqrt({(((a + c) ^ 2) + ((b + d) ^ 2) + ((g + f) ^ 2))}))", "checkingType": "absdiff", "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "answerSimplification": "std", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}, {"checkVariableNames": false, "type": "jme", "vsetRange": [0, 1], "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "answer": "({(b + d)} / Sqrt({(((a + c) ^ 2) + ((b + d) ^ 2) + ((g + f) ^ 2))}))", "checkingType": "absdiff", "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "answerSimplification": "std", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}, {"checkVariableNames": false, "type": "jme", "vsetRange": [0, 1], "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "answer": "({(g + f)} / Sqrt({(((a + c) ^ 2) + ((b + d) ^ 2) + ((g + f) ^ 2))}))", "checkingType": "absdiff", "showPreview": true, "expectedVariableNames": [], "failureRate": 1, "vsetRangePoints": 5, "checkingAccuracy": 0.001, "answerSimplification": "std", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Let $\\mathbf{z}=\\mathbf{v}+\\mathbf{w}$.</p>\n<p>Calculate&nbsp;the unit vector $\\mathbf{\\hat{z}}$ in the direction of $\\mathbf{z}$. Write $\\mathbf{\\hat{z}}$ as a row vector.</p>\n<p>$\\mathbf{\\hat{z}}= \\big($ [[0]],&nbsp;[[1]],&nbsp;[[2]] $\\big)$</p>\n<p>You must enter your answers exactly, using the function <code>sqrt(x)</code>&nbsp;as necessary.</p>", "scripts": {}}, {"type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "customMarkingAlgorithm": "", "sortAnswers": false, "gaps": [{"type": "matrix", "allowResize": false, "allowFractions": false, "correctAnswerFractions": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "numColumns": 1, "variableReplacements": [], "customMarkingAlgorithm": "", "correctAnswer": "a4*v1", "tolerance": 0, "markPerCell": false, "numRows": "3", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}, {"type": "matrix", "allowResize": false, "allowFractions": false, "correctAnswerFractions": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "numColumns": 1, "variableReplacements": [], "customMarkingAlgorithm": "", "correctAnswer": "b4*v2", "tolerance": 0, "markPerCell": false, "numRows": "3", "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "scripts": {}}], "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "prompt": "<p>Calculate</p>\n<p>$\\var{a4}\\mathbf{v} = $ [[0]]</p>\n<p>$\\var{b4}\\mathbf{w} = $ [[1]]</p>", "scripts": {}}], "advice": "<h4>a)</h4>\n<p>\\[\\boldsymbol{v}+\\boldsymbol{w} = \\var{vector(a,b,g)} + \\var{vector(c,d,f)} = \\var{vector(a+c,b+d,g+f)} \\]</p>\n<h4>b)</h4>\n<p>In general for a vector $\\boldsymbol{x}= \\begin{pmatrix}x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, we have $\\lVert \\boldsymbol{x} \\rVert = \\sqrt{x_1^2+x_2^2+x_3^2}$.</p>\n<p>Hence:</p>\n<p>\\begin{align}<br/>\\lVert \\boldsymbol{v} \\rVert &amp;= \\sqrt{\\var{a^2}+\\var{b^2}+\\var{g^2}} = \\simplify[all]{ sqrt({a^2+b^2+g^2})} \\\\<br/>\\lVert \\boldsymbol{w} \\rVert &amp;= \\sqrt{\\var{c^2}+\\var{d^2}+\\var{f^2}} = \\simplify[all]{ sqrt({c^2+d^2+f^2})} \\\\ <br/>\\lVert \\boldsymbol{v+w} \\rVert &amp;= \\sqrt{\\var{(a+c)^2}+\\var{(b+d)^2}+\\var{(g+f)^2}} = \\simplify[all]{ sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>Given a vector $\\boldsymbol{x}= \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, the unit vector parallel to $\\boldsymbol{x}$ is given by:</p>\n<p>\\[ \\boldsymbol{u_x}&nbsp;= \\frac{1}{\\lVert \\boldsymbol{x} \\rVert} \\begin{pmatrix} x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix} = \\begin{pmatrix} \\frac{x_1}{\\lVert \\boldsymbol{x} \\rVert} \\\\ \\frac{x_2}{\\lVert \\boldsymbol{x} \\rVert} \\\\ \\frac{x_3}{\\lVert \\boldsymbol{x} \\rVert}&nbsp;\\end{pmatrix} \\]</p>\n<p>For this example we have $\\lVert \\boldsymbol{v+w} \\rVert =\\simplify[std]{sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}$, hence:</p>\n<p>\\begin{align} <br/>&amp;&amp;\\boldsymbol{z} = \\boldsymbol{v} + \\boldsymbol{w} &amp;= \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\<br/>\\implies &amp;&amp; \\boldsymbol{u_z}&nbsp;&amp;= \\frac{1}{\\sqrt{\\var{ssquares}}} \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\[1em]&nbsp;<br/>&amp;&amp; &amp;= \\begin{pmatrix} \\simplify[std]{{a+c}/sqrt({ssquares})} \\\\ \\simplify[std]{{b+d}/sqrt({ssquares})} \\\\ \\simplify[std]{{g+f}/sqrt({ssquares})} \\end{pmatrix}<br/>\\end{align}</p>\n<h4>d)</h4>\n<p>\\begin{align}<br/>\\var{a4}\\boldsymbol{v} &amp;= \\simplify{vector({a4}*{a}, {a4}*{b}, {a4}*{g})} \\\\[1em]<br/>&amp;= \\var{a4*vector(a,b,g)}<br/>\\end{align}</p>\n<p>\\begin{align}<br/>\\var{-b4}\\boldsymbol{v} &amp;= \\simplify{vector({-b4}*{c}, {-b4}*{d}, {-b4}*{f})} \\\\[1em]<br/>&amp;= \\var{-b4*vector(c,d,f)}<br/>\\end{align}</p>\n<h4>e)</h4>\n<p>Using the information above, the unit vector parallel to $\\boldsymbol{v}$ is:</p>\n<p>\\[ \\boldsymbol{u_v}&nbsp;= \\begin{pmatrix} \\simplify[std]{{a}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{b}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{g}/sqrt({ssquaresA})}&nbsp;\\end{pmatrix}&nbsp;\\]</p>\n<p>and the unit vector anti-parallel to $\\boldsymbol{w}$ is:</p>\n<p>\\[ -\\boldsymbol{u_w} = \\begin{pmatrix} \\simplify[std]{{-c}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-d}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-f}/sqrt({ssquaresB})}&nbsp;\\end{pmatrix}&nbsp;\\]</p>", "variables": {"b": {"templateType": "anything", "definition": "s2*random(2..9)", "name": "b", "description": "", "group": "Ungrouped variables"}, "q": {"templateType": "anything", "definition": "M+N", "name": "q", "description": "", "group": "Ungrouped variables"}, "v": {"templateType": "anything", "definition": "v1+v2", "name": "v", "description": "", "group": "Ungrouped variables"}, "s1": {"templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": "", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "definition": "s4*random(2..9)", "name": "d", "description": "", "group": "Ungrouped variables"}, "g": {"templateType": "anything", "definition": "s1*random(2..9)", "name": "g", "description": "", "group": "Ungrouped variables"}, "a4": {"templateType": "anything", "definition": "random(3..9)", "name": "a4", "description": "", "group": "Ungrouped variables"}, "m": {"templateType": "anything", "definition": "matrix([a,b],[c,d])", "name": "m", "description": "", "group": "Ungrouped variables"}, "s5": {"templateType": "anything", "definition": "random(1,-1)", "name": "s5", "description": "", "group": "Ungrouped variables"}, "a": {"templateType": "anything", "definition": "s1*random(2..9)", "name": "a", "description": "", "group": "Ungrouped variables"}, "f": {"templateType": "anything", "definition": "random(2..9)", "name": "f", "description": "", "group": "Ungrouped variables"}, "s3": {"templateType": "anything", "definition": "random(1,-1)", "name": "s3", "description": "", "group": "Ungrouped variables"}, "v2": {"templateType": "anything", "definition": "vector(c,d,f)", "name": "v2", "description": "", "group": "Ungrouped variables"}, "ssquares": {"templateType": "anything", "definition": "(a+c)^2+(b+d)^2+(f+g)^2", "name": "ssquares", "description": "", "group": "Ungrouped variables"}, "v1": {"templateType": "anything", "definition": "vector(a,b,g)", "name": "v1", "description": "", "group": "Ungrouped variables"}, "s4": {"templateType": "anything", "definition": "random(1,-1)", "name": "s4", "description": "", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "definition": "s3*random(2..9)", "name": "c", "description": "", "group": "Ungrouped variables"}, "b4": {"templateType": "anything", "definition": "-random(3..9)", "name": "b4", "description": "", "group": "Ungrouped variables"}, "ssquaresa": {"templateType": "anything", "definition": "(a)^2+(b)^2+(g)^2", "name": "ssquaresa", "description": "", "group": "Ungrouped variables"}, "n": {"templateType": "anything", "definition": "matrix([a,b],[c,d])", "name": "n", "description": "", "group": "Ungrouped variables"}, "s2": {"templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": "", "group": "Ungrouped variables"}, "ssquaresb": {"templateType": "anything", "definition": "(c)^2+(d)^2+(f)^2", "name": "ssquaresb", "description": "", "group": "Ungrouped variables"}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Elementary operations on vectors; sum, modulus, unit vector, scalar multiple.&nbsp;</p>"}, "variable_groups": [], "statement": "<p>You are given the vectors</p>\n<p>\\begin{align}<br/> \\mathbf{v} &amp; =\\simplify[std]{vector({a},{b},{g})}, &amp;<br/>\\mathbf{w} &amp;= \\simplify[std]{vector({c},{d},{f})}\\qquad \\in{\\mathbb R}^3.<br/>\\end{align}</p>\n<p>Enter your answers to the following questions exactly, using the function <code>sqrt(x)</code> if necessary.</p>", "preamble": {"js": "", "css": ""}, "type": "question"}, {"name": "Dot product and angle between two vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}], "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"gaps": [{"unitTests": [], "mustBeReducedPC": 0, "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "ans1-0.005", "maxValue": "ans1+0.005", "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 1, "variableReplacements": []}, {"unitTests": [], "mustBeReducedPC": 0, "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "ansrad-0.05", "maxValue": "ansrad+0.05", "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 1, "variableReplacements": []}, {"unitTests": [], "mustBeReducedPC": 0, "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "dot_of_ab-0.001", "maxValue": "dot_of_ab+0.001", "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 1, "variableReplacements": []}], "unitTests": [], "type": "gapfill", "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>$\\mathbf{a}=\\pmatrix{\\var{a[0]},\\var{a[1]},\\var{a[2]}}$ and $\\mathbf{ b}=\\pmatrix{\\var{b[0]},\\var{b[1]},\\var{b[2]}}$</p>\n<p>$\\mathbf{a} \\cdot \\mathbf{b}=$ [[2]]</p>\n<p>$\\cos({\\theta})=$ [[0]]&nbsp; (<span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">Give your answer</span> to two decimal places)</p>\n<p><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">$\\theta=$ [[1]]&nbsp;</span><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">(Give your answer, in radians to one decimal place )</span></p>", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 0, "variableReplacements": []}, {"gaps": [{"unitTests": [], "mustBeReducedPC": 0, "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "ans2-0.005", "maxValue": "ans2+0.005", "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 1, "variableReplacements": []}, {"unitTests": [], "mustBeReducedPC": 0, "type": "numberentry", "scripts": {}, "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "mustBeReduced": false, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "minValue": "ansrad2-0.05", "maxValue": "ansrad2+0.05", "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 1, "variableReplacements": []}, {"answer": "{dot_of_cd}", "unitTests": [], "type": "jme", "scripts": {}, "vsetRange": [0, 1], "failureRate": 1, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showPreview": true, "checkingAccuracy": 0.001, "checkingType": "absdiff", "checkVariableNames": false, "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 1, "expectedVariableNames": [], "variableReplacements": []}], "unitTests": [], "type": "gapfill", "sortAnswers": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p style=\"background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; min-height: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">&nbsp;$ \\mathbf{c}=\\var{c[0]}i&nbsp; + \\var{c[1]}j + \\var{c[2]}k$ and $\\mathbf{d}= \\var{d[0]}i+ \\var{d[1]}j+\\var{d[2]}k$</p>\n<p style=\"background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; min-height: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">$\\mathbf{c} \\cdot \\mathbf{d}=$ [[2]]</p>\n<p style=\"background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; min-height: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">$\\cos({\\theta})=$ [[0]]&nbsp; (Give your answer to two decimal places)</p>\n<p style=\"background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; min-height: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">$\\theta=$ [[1]]</span><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">&nbsp; (Give your answer, in radians to one decimal place)</span></p>", "customMarkingAlgorithm": "", "showCorrectAnswer": true, "marks": 0, "variableReplacements": []}], "functions": {}, "variables": {"ans1": {"name": "ans1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "precround(dot(a,b)/(lena*lenb),2)"}, "a": {"name": "a", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))"}, "d": {"name": "d", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "vector(repeat(random(2..9)*sign(random(1,1)),3))"}, "ansrad2": {"name": "ansrad2", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "precround(arccos(ans2),1)"}, "dot_of_cd": {"name": "dot_of_cd", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "dot(c,d)"}, "dot_of_ab": {"name": "dot_of_ab", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "dot(a,b)"}, "ansrad": {"name": "ansrad", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "precround(arccos(ans1),1)"}, "lena": {"name": "lena", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "abs(a)"}, "lenb": {"name": "lenb", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "abs(b)"}, "lend": {"name": "lend", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "abs(d)"}, "c": {"name": "c", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "vector(repeat(random(2..9)*sign(random(1,1)),3))"}, "b": {"name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))"}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "precround(dot(c,d)/(lenc*lend),2)"}, "lenc": {"name": "lenc", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "abs(c)"}}, "advice": "<p><strong>Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.</strong></p>\n<p>The dot product of two vectors $\\boldsymbol{a}=\\pmatrix{a_1,a_2,a_3}$ and $\\boldsymbol{b}=\\pmatrix{b_1,b_2,b_3}$ is given by</p>\n<p>\\[\\boldsymbol{a\\cdot b}=a_1b_1+a_2b_2+a_3b_3\\] </p>\n<p></p>\n<p>$\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2}$ ,$\\lvert\\boldsymbol{b}\\rvert=\\sqrt{b_1^2+b_2^2+b_3^2}$ are the lengths of the vectors $\\boldsymbol{a}$ and $\\boldsymbol{b}$.</p>\n<p></p>\n<p>and so</p>\n<p>\\[\\cos(\\theta)=\\frac{\\boldsymbol{a\\cdot b}}{\\lvert\\boldsymbol{a}\\rvert \\lvert\\boldsymbol{b}\\rvert}=\\frac{a_1b_1+a_2b_2+a_3b_3}{\\lvert\\boldsymbol{a}\\rvert \\lvert\\boldsymbol{b}\\rvert}.\\]</p>\n<p>In part a) therefore, we have</p>\n<p>\\[\\cos(\\theta)=\\frac{\\var{dot(a,b)}}{\\var{precround(lena,2)}\\times\\var{precround(lenb,2)}}=\\frac{\\var{dot(a,b)}}{\\var{precround(lena*lenb,2)}}=\\var{ans1} \\; \\text{to 2d.p.,}\\]</p>\n<p>Which gives an angle $\\theta =\\var{ansrad}$ radians to 1 d.p.</p>", "metadata": {"description": "<p>Find the dot product and the angle between two vectors</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "tags": [], "statement": "<p>Find the angle&nbsp; $\\theta$&nbsp; between the following pairs of vectors.</p>", "ungrouped_variables": ["a", "lenb", "c", "b", "lenc", "d", "lend", "ans1", "ans2", "lena", "ansrad", "ansrad2", "dot_of_ab", "dot_of_cd"], "type": "question"}, {"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/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}], "parts": [{"variableReplacements": [], "showCorrectAnswer": true, "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 <g class=\"gr_ gr_6 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling ins-del multiReplace\" id=\"6\" data-gr-id=\"6\">mulptiple</g> of $\\pi$. You input $\\pi$ as pi.</p>", "marks": 0, "unitTests": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "sortAnswers": false, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacements": [], "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "unitTests": [], "scripts": {}, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{angle}", "customMarkingAlgorithm": "", "type": "numberentry", "variableReplacementStrategy": "originalfirst", "minValue": "{angle}", "showFeedbackIcon": true}, {"variableReplacements": [], "showCorrectAnswer": true, "marks": 1, "unitTests": [], "scripts": {}, "checkingType": "absdiff", "expectedVariableNames": [], "answer": "{precround(angle*pi/180,4)}", "vsetRange": [0, 1], "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.0001, "customMarkingAlgorithm": "", "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showPreview": true, "checkVariableNames": false, "failureRate": 1, "showFeedbackIcon": true}], "showFeedbackIcon": true}], "functions": {}, "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>", "statement": "<p>Given the vectors<br/>$\\mathbf{a}=\\var{fa}\\mathbf{i}+\\var{sa}\\mathbf{j}+\\var{ta}\\mathbf{k},\\;\\;\\;\\mathbf{b}=\\var{fb}\\mathbf{i}+\\var{sb}\\mathbf{j}+ \\var{tb}\\mathbf{k}$</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>", "tags": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["s1", "a", "s4", "fa", "angle", "g", "fb", "tb", "sb", "ta", "c", "sa", "s2", "s3", "u", "d", "b", "t"], "variables": {"a": {"definition": "if(t=1,2,1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a"}, "s4": {"definition": "if(s1=s3 ,-s2,random(-1,1))", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s4"}, "fa": {"definition": "if(t=1,0,s1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "fa"}, "angle": {"definition": "precround(180/pi*arccos(g/2),1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "angle"}, "s1": {"definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s1"}, "fb": {"definition": "if(u=1,0,s3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "fb"}, "tb": {"definition": "if(u=3,0,s4)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "tb"}, "sb": {"definition": "if(u=2,0,if(u=1,s3,s4))", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "sb"}, "ta": {"definition": "if(t=3,0,s2)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ta"}, "c": {"definition": "if(u=1,2,1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c"}, "sa": {"definition": "if(t=2,0,if(t=1,s1,s2))", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "sa"}, "s2": {"definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s2"}, "s3": {"definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s3"}, "u": {"definition": "random(1,2,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "u"}, "d": {"definition": "if(u=3,2,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d"}, "b": {"definition": "if(t=3,2,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b"}, "g": {"definition": "{fa*fb+sa*sb+ta*tb}", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "g"}, "t": {"definition": "random(1,2,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "t"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the angle between two vectors</p>"}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variable_groups": [], "type": "question"}, {"name": "Cross product", "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": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}], "statement": "<p>Given the vectors:<br/>\\[{\\bf a}=\\simplify{{a} v:i+{b}v:j+{g}v:k},\\;\\;\\;{\\bf b}=\\simplify[std]{{c}v:i+{d}v:j+{f}v:k}\\]</p>\n<p>answer the following question:</p>", "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"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"s4": {"group": "Ungrouped variables", "description": "", "definition": "random(1,-1)", "name": "s4", "templateType": "anything"}, "f": {"group": "Ungrouped variables", "description": "", "definition": "random(2..9)", "name": "f", "templateType": "anything"}, "c": {"group": "Ungrouped variables", "description": "", "definition": "s3*random(2..9)", "name": "c", "templateType": "anything"}, "inner": {"group": "Ungrouped variables", "description": "", "definition": "{a*c+b*d+f*g}", "name": "inner", "templateType": "anything"}, "g": {"group": "Ungrouped variables", "description": "", "definition": "s1*random(2..9)", "name": "g", "templateType": "anything"}, "s2": {"group": "Ungrouped variables", "description": "", "definition": "random(1,-1)", "name": "s2", "templateType": "anything"}, "a": {"group": "Ungrouped variables", "description": "", "definition": "s1*random(2..9)", "name": "a", "templateType": "anything"}, "d": {"group": "Ungrouped variables", "description": "", "definition": "s4*random(2..9)", "name": "d", "templateType": "anything"}, "b": {"group": "Ungrouped variables", "description": "", "definition": "s2*random(2..9)", "name": "b", "templateType": "anything"}, "s1": {"group": "Ungrouped variables", "description": "", "definition": "random(1,-1)", "name": "s1", "templateType": "anything"}, "s3": {"group": "Ungrouped variables", "description": "", "definition": "random(1,-1)", "name": "s3", "templateType": "anything"}, "s5": {"group": "Ungrouped variables", "description": "", "definition": "random(1,-1)", "name": "s5", "templateType": "anything"}}, "functions": {}, "tags": [], "parts": [{"type": "gapfill", "scripts": {}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "gaps": [{"type": "numberentry", "scripts": {}, "extendBaseMarkingAlgorithm": true, "maxValue": "{b*f-g*d}", "showFeedbackIcon": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{b*f-g*d}", "allowFractions": false, "marks": 6, "mustBeReducedPC": 0, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "showCorrectAnswer": true}, {"type": "numberentry", "scripts": {}, "extendBaseMarkingAlgorithm": true, "maxValue": "{g*c-a*f}", "showFeedbackIcon": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{g*c-a*f}", "allowFractions": false, "marks": 6, "mustBeReducedPC": 0, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "showCorrectAnswer": true}, {"type": "numberentry", "scripts": {}, "extendBaseMarkingAlgorithm": true, "maxValue": "{a*d-b*c}", "showFeedbackIcon": true, "correctAnswerFraction": false, "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a*d-b*c}", "allowFractions": false, "marks": 6, "mustBeReducedPC": 0, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "showCorrectAnswer": true}], "customMarkingAlgorithm": "", "prompt": "<p>Find ${{\\bf a}\\times {\\bf b}} =\\;\\;$ [[0]]${\\bf i}+ $[[1]]${\\bf j}+ $[[2]]${\\bf k}$</p>", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "marks": 0, "variableReplacements": [], "unitTests": [], "showCorrectAnswer": true}], "advice": "<p>\\[ \\begin{eqnarray*} \\boldsymbol{a\\times b}&amp;=&amp; \\begin{vmatrix} \\boldsymbol{i} &amp; \\boldsymbol{j} &amp;\\boldsymbol{k}\\\\ \\var{a} &amp; \\var{b} &amp; \\var{g}\\\\ \\var{c} &amp; \\var{d} &amp; \\var{f} \\end{vmatrix}\\\\ \\\\ &amp;=&amp;\\simplify[]{({b}*{f}-{g}*{d})v:i + ({g}*{c} - {a}*{f})v:j+({a}*{d}-{b}*{c})v:k}\\\\ \\\\ &amp;=&amp;\\simplify[std]{{b*f-g*d}v:i+{g*c-a*f}v:j+{a*d-b*c}v:k} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "inner"], "type": "question"}, {"name": " Vectors: when perpendicular 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}], "statement": "<p>Given the vectors:<br/>\\[\\mathbf{a}=\\simplify[std]{{a}v:i+{b}v:j+lambda*v:k},\\;\\;\\;\\mathbf{b}=\\simplify[std]{{c}v:i+{d}v:j+{f}v:k}\\]</p>", "variable_groups": [], "functions": {}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4"], "tags": [], "advice": "<h4>a)</h4>\n<p>$\\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>", "parts": [{"sortAnswers": false, "gaps": [{"vsetRangePoints": 5, "showPreview": true, "expectedVariableNames": [], "variableReplacements": [], "type": "jme", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "failureRate": 1, "variableReplacementStrategy": "originalfirst", "answerSimplification": "std", "checkVariableNames": false, "vsetRange": [0, 1], "unitTests": [], "checkingAccuracy": 0.001, "notallowed": {"message": "<p>Enter as a fraction or an integer and not as a decimal.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "marks": 1.5, "checkingType": "absdiff", "answer": "{-a*c-b*d}/{f}", "scripts": {}}], "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "unitTests": [], "extendBaseMarkingAlgorithm": true, "prompt": "<p>Find $\\lambda$ such that $\\mathbf{a}$ and $\\mathbf{b}$ 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>", "showFeedbackIcon": true, "marks": 0, "scripts": {}}, {"sortAnswers": false, "gaps": [{"mustBeReduced": false, "maxValue": "{0}", "variableReplacements": [], "type": "numberentry", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "unitTests": [], "correctAnswerFraction": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "marks": 0.5, "mustBeReducedPC": 0, "minValue": "{0}", "allowFractions": false, "scripts": {}, "notationStyles": ["plain", "en", "si-en"]}], "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "unitTests": [], "extendBaseMarkingAlgorithm": true, "prompt": "<p>Find $\\lambda$ such that $\\mathbf{a}$ is in the $xy$ plane:</p>\n<p>$\\lambda =\\;\\;$ [[0]].</p>", "showFeedbackIcon": true, "marks": 0, "scripts": {}}], "metadata": {"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 $\\boldsymbol{A,\\<g class=\"gr_ gr_3 gr-alert gr_gramm gr_inline_cards gr_run_anim Style replaceWithoutSep\" id=\"3\" data-gr-id=\"3\">;B}</g>$&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"}, "preamble": {"js": "", "css": ""}, "variables": {"s3": {"name": "s3", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}, "c": {"name": "c", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s3*random(2..9)"}, "s1": {"name": "s1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}, "a": {"name": "a", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s1*random(2..9)"}, "g": {"name": "g", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s1*random(2..9)"}, "d": {"name": "d", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s4*random(2..9)"}, "b": {"name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s2*random(2..9)"}, "f": {"name": "f", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(2,4,5,10)"}, "s5": {"name": "s5", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}, "s4": {"name": "s4", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}, "s2": {"name": "s2", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question"}]}], "name": "Vectors - Ex. Sheet 2 ", "showQuestionGroupNames": false, "percentPass": "0", "showstudentname": true, "feedback": {"intro": "", "feedbackmessages": [], "showanswerstate": true, "allowrevealanswer": true, "showactualmark": true, "advicethreshold": 0, "showtotalmark": true}, "timing": {"timeout": {"action": "none", "message": ""}, "allowPause": true, "timedwarning": {"action": "none", "message": ""}}, "navigation": {"showfrontpage": false, "reverse": true, "allowregen": true, "preventleave": true, "showresultspage": "oncompletion", "browse": true, "onleave": {"action": "none", "message": ""}}, "duration": 0, "type": "exam", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}], "extensions": [], "custom_part_types": [], "resources": []}