// Numbas version: exam_results_page_options {"name": "Determine if vectors are perpendicular", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"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"], "name": "Determine if vectors are perpendicular", "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

When are vectors $\\boldsymbol{v,\\;w}$ orthogonal?

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Part b) is not answered in Advice, the given solution is for a different question.

"}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find $\\lambda \\in \\mathbb{R}$ such that $\\boldsymbol{v}$ and $\\boldsymbol{w}$ are orthogonal.

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$\\lambda = $ [[0]]

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Find $\\lambda \\in \\mathbb{R}$ such that the vector $\\boldsymbol{u} = \\simplify[fractionnumbers]{{u}}$ is contained in the plane through the origin parallel to $\\boldsymbol{v}$ and $\\boldsymbol{w}$.

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$\\lambda =$ [[0]]

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You are given the vectors $\\boldsymbol{v} = \\begin{pmatrix}\\var{a} \\\\ \\var{b} \\\\ \\lambda \\end{pmatrix}$ and $\\boldsymbol{w} = \\begin{pmatrix} \\var{c} \\\\ \\var{d} \\\\ \\var{f} \\end{pmatrix}$.

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Enter your answers to the following questions as fractions or integers, not decimals.

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a)

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$\\boldsymbol{v}$ and $\\boldsymbol{w}$ are perpendicular to one another when $\\boldsymbol{v} \\cdot \\boldsymbol{w} = 0$.

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Now

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\\begin{align}
\\boldsymbol{v} \\cdot \\boldsymbol{w} &= \\simplify[]{{a}*{c}+{b}*{d}+lambda*{f}} \\\\
&= \\simplify[std]{{f}*lambda+{a*c+b*d}}
\\end{align}

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Hence

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\\[\\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}}\\]

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b)

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

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