// Numbas version: exam_results_page_options {"percentPass": 0, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Vectors 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}], "ungrouped_variables": ["t", "u", "a", "b", "c", "d", "fa", "fb", "g", "sa", "sb", "ta", "tb"], "variable_groups": [{"name": "Initial vectors", "variables": ["s1", "s2", "s3", "s4", "units", "direction_v", "direction_w", "v", "w"]}, {"name": "Result", "variables": ["angle", "precision"]}], "advice": "

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.

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Here

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\\begin{align}
\\lVert \\boldsymbol{v} \\rVert &= \\simplify[]{sqrt({s1}^2 + {s2}^2)} \\\\

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#### b)

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$\\boldsymbol{v}$ is in the $xy$ plane when $\\lambda=0$.

", "variablesTest": {"maxRuns": 100, "condition": "u<>vector(0,0,0)"}, "functions": {}, "variable_groups": [], "type": "question"}, {"name": "Vectors 3 (dot and cross)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}], "ungrouped_variables": [], "variable_groups": [], "advice": "

1. $\\boldsymbol{(v\\cdot w)\\cdot u}$ is undefined as $\\boldsymbol{v\\cdot w}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{u}$.

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2. $\\boldsymbol{(v\\cdot w) u}$ is a vector and is a scalar multiple of $\\boldsymbol{u}$ as $\\boldsymbol{v \\cdot w}$ is a scalar.

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3. $\\boldsymbol{(v \\cdot w)\\times u}$ is undefined as $\\boldsymbol{v\\cdot w}$ is a scalar and the cross product is only defined between vectors.

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4. $\\boldsymbol{(v\\times w)\\times u}$ is a vector as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the cross product between vectors produces a vector.

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5. $\\boldsymbol{(v\\times w)\\cdot u}$ is a scalar as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the inner or dot product is between vectors and produces a scalar.

", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "metadata": {"description": "

Determine if various combinations of vectors are defined or not.

", "notes": "\n \t\t

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16/07/2012:

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\n \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

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.

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In this question, the symbol $\\cdot$ denotes the inner product and $\\times$ always denotes the cross product.

", "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"shuffleAnswers": true, "shuffleChoices": true, "layout": {"type": "all", "expression": ""}, "type": "m_n_x", "answers": ["

Scalar

", "

Vector

", "

Undefined

"], "choices": ["$\\boldsymbol{(v\\cdot w)\\cdot u}$", "$\\boldsymbol{(v\\cdot w)u}$", "$\\boldsymbol{(v\\cdot w)\\times u}$", "$\\boldsymbol{(v\\times w)\\times u}$", "

$\\boldsymbol{(v\\times w)\\cdot u}$

"], "maxAnswers": 0, "variableReplacementStrategy": "originalfirst", "matrix": [[0, 0, 0.4], [0, 0.4, 0], [0, 0, 0.4], [0, 0.4, 0], [0.4, 0, 0]], "scripts": {}, "minAnswers": 0, "showCorrectAnswer": true, "warningType": "none", "minMarks": 0, "maxMarks": 0, "displayType": "radiogroup", "variableReplacements": [], "marks": 0}], "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "variables": {}, "tags": ["checked2015", "cross product", "dot product", "inner product", "MAS1602", "mas1602", "scalar product", "scalars", "unused", "vector", "Vector", "vector product", "vectors"], "functions": {}, "preamble": {"css": "", "js": ""}, "showQuestionGroupNames": false, "type": "question"}, {"name": "Vectors 4 (dot product)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}], "type": "question", "preamble": {"js": "", "css": ""}, "parts": [{"gaps": [{"type": "numberentry", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showPrecisionHint": false, "scripts": {}, "marks": 2, "allowFractions": false, "minValue": "{inner}", "correctAnswerFraction": false, "showCorrectAnswer": true, "maxValue": "{inner}"}], "type": "gapfill", "variableReplacements": [], "scripts": {}, "marks": 0, "prompt": "

Find $\\boldsymbol{v \\cdot w} =$ [[0]]

", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "metadata": {"description": "

Given vectors $\\boldsymbol{v}$ and $\\boldsymbol{w}$, find their inner product.

", "licence": "Creative Commons Attribution 4.0 International", "notes": "\n \t\t

15/07/2012:

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16/07/2012:

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Question appears to be working correctly.

\n \t\t"}, "tags": ["checked2015", "dot product", "dot product of two vectors", "inner product", "mas1602", "MAS1602", "scalar product", "three dimensional vectors", "unused", "vectors"], "advice": "

\\begin{align}
\\boldsymbol{v \\cdot w} &= \\var{vector(a,b,g)} \\boldsymbol{\\cdot} \\var{vector(c,d,f)} \\\\
&= \\simplify[]{{a}*{c}+{b}*{d}+{g}*{f}} \\\\
&= \\var{inner}
\\end{align}

", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variable_groups": [], "variables": {"b": {"definition": "s2*random(2..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "b"}, "d": {"definition": "s4*random(2..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "d"}, "inner": {"definition": "{a*c+b*d+f*g}", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "inner"}, "g": {"definition": "s1*random(2..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "g"}, "c": {"definition": "s3*random(2..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "c"}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "s2"}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "s5"}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "s3"}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "s4"}, "f": {"definition": "random(2..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "f"}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "s1"}, "a": {"definition": "s1*random(2..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "a"}}, "showQuestionGroupNames": false, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "inner"], "variablesTest": {"maxRuns": 100, "condition": ""}, "question_groups": [{"pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered", "name": ""}], "statement": "

You are given the vectors $\\boldsymbol{v}= \\var{vector(a,b,g)}$ and $\\boldsymbol{w} = \\var{vector(c,d,f)}$ in $\\mathbb{R}^3$.

", "functions": {}}, {"name": "Vector 5 (cross product)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}], "showQuestionGroupNames": false, "statement": "

You are given the vectors $\\boldsymbol{v} = \\var{vector(a,b,g)}$, $\\boldsymbol{w} = \\var{vector(c,d,f)}$.

", "question_groups": [{"name": "", "questions": [], "pickQuestions": 0, "pickingStrategy": "all-ordered"}], "functions": {}, "tags": ["3 dimensional vector", "checked2015", "cross product", "three dimensional vectors", "unused", "Vector", "vector", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "result", "s1", "s2", "s3", "s4", "s5"], "parts": [{"type": "gapfill", "gaps": [{"scripts": {}, "correctAnswerFractions": false, "marks": "3", "variableReplacements": [], "type": "matrix", "markPerCell": false, "showCorrectAnswer": true, "allowResize": false, "correctAnswer": "result", "allowFractions": false, "numColumns": 1, "tolerance": 0, "numRows": "3", "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "scripts": {}, "prompt": "

Find

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$\\boldsymbol{v} \\times \\boldsymbol{w} =$ [[0]]

", "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "metadata": {"notes": "

14/7/2015

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16/07/2012:

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Question appears to be working correctly.

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", "description": "

Given vectors $\\boldsymbol{A,\\;B}$, find $\\boldsymbol{A\\times B}$

", "licence": "Creative Commons Attribution 4.0 International"}, "advice": "

\\begin{align}
\\boldsymbol{v} \\times \\boldsymbol{w} &= \\begin{pmatrix} \\simplify[basic]{{b}*{f}-{g}*{d}} \\\\ \\simplify[basic]{{g}*{c}-{a}*{f}} \\\\ \\simplify[basic]{{a}*{d}-{b}*{c}}  \\end{pmatrix} \\\\[1em]
&= \\var{result}
\\end{align}

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