// Numbas version: exam_results_page_options {"name": "MATH6005 Assessment 2_Q2of6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "functions": {}, "metadata": {"description": "

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple.

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

a)

\\[\\boldsymbol{v}+\\boldsymbol{w} = \\var{vector(a,b,g)} + \\var{vector(c,d,f)} = \\var{vector(a+c,b+d,g+f)} \\]

b)

In general for a vector $\\boldsymbol{x}= \\begin{pmatrix}x_1 \\\\ x_2 \\\\ x_3 \\end{pmatrix}$, we have $\\lVert \\boldsymbol{x} \\rVert = \\sqrt{x_1^2+x_2^2+x_3^2}$.

Hence:

\\begin{align}
\\lVert \\boldsymbol{v} \\rVert &= \\sqrt{\\var{a^2}+\\var{b^2}+\\var{g^2}} = \\simplify[all]{ sqrt({a^2+b^2+g^2})} \\\\
\\lVert \\boldsymbol{w} \\rVert &= \\sqrt{\\var{c^2}+\\var{d^2}+\\var{f^2}} = \\simplify[all]{ sqrt({c^2+d^2+f^2})} \\\\
\\lVert \\boldsymbol{v+w} \\rVert &= \\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})}
\\end{align}

c)

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:

\\[ \\boldsymbol{u_x} = \\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} \\end{pmatrix} \\]

For this example we have $\\lVert \\boldsymbol{v+w} \\rVert =\\simplify[std]{sqrt({(a+c)^2+(b+d)^2+(f+g)^2})}$, hence:

\\begin{align}
&&\\boldsymbol{z} = \\boldsymbol{v} + \\boldsymbol{w} &= \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\
\\implies && \\boldsymbol{u_z} &= \\frac{1}{\\sqrt{\\var{ssquares}}} \\begin{pmatrix} \\var{a+c} \\\\ \\var{b+d} \\\\ \\var{g+f} \\end{pmatrix} \\\\[1em]
&& &= \\begin{pmatrix} \\simplify[std]{{a+c}/sqrt({ssquares})} \\\\ \\simplify[std]{{b+d}/sqrt({ssquares})} \\\\ \\simplify[std]{{g+f}/sqrt({ssquares})} \\end{pmatrix}
\\end{align}

d)

\\begin{align}
\\var{a4}\\boldsymbol{v} &= \\simplify{vector({a4}*{a}, {a4}*{b}, {a4}*{g})} \\\\[1em]
&= \\var{a4*vector(a,b,g)}
\\end{align}

\\begin{align}
\\var{-b4}\\boldsymbol{v} &= \\simplify{vector({-b4}*{c}, {-b4}*{d}, {-b4}*{f})} \\\\[1em]
&= \\var{-b4*vector(c,d,f)}
\\end{align}

e)

Using the information above, the unit vector parallel to $\\boldsymbol{v}$ is:

\\[ \\boldsymbol{u_v} = \\begin{pmatrix} \\simplify[std]{{a}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{b}/sqrt({ssquaresA})} \\\\ \\simplify[std]{{g}/sqrt({ssquaresA})} \\end{pmatrix} \\]

and the unit vector anti-parallel to $\\boldsymbol{w}$ is:

\\[ -\\boldsymbol{u_w} = \\begin{pmatrix} \\simplify[std]{{-c}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-d}/sqrt({ssquaresB})} \\\\ \\simplify[std]{{-f}/sqrt({ssquaresB})} \\end{pmatrix} \\]

", "statement": "

You are given the vectors

\\begin{align}
\\mathbf{v} & =\\simplify[std]{vector({a},{b},{g})}, &
\\mathbf{w} &= \\simplify[std]{vector({c},{d},{f})}\\qquad \\in{\\mathbb R}^3.
\\end{align}

Enter your answers to the following questions exactly, using the function sqrt(x) if necessary.

", "variable_groups": [], "name": "MATH6005 Assessment 2_Q2of6", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "tags": [], "variables": {"ssquaresa": {"name": "ssquaresa", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a)^2+(b)^2+(g)^2"}, "v2": {"name": "v2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "vector(c,d,f)"}, "a": {"name": "a", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)"}, "d": {"name": "d", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(2..9)"}, "ssquaresb": {"name": "ssquaresb", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "(c)^2+(d)^2+(f)^2"}, "s4": {"name": "s4", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "a4": {"name": "a4", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)"}, "v1": {"name": "v1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "vector(a,b,g)"}, "n": {"name": "n", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([a,b],[c,d])"}, "v": {"name": "v", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "v1+v2"}, "m": {"name": "m", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "matrix([a,b],[c,d])"}, "b4": {"name": "b4", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "-random(3..9)"}, "s2": {"name": "s2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "c": {"name": "c", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..9)"}, "s3": {"name": "s3", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "s5": {"name": "s5", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "f": {"name": "f", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)"}, "g": {"name": "g", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)"}, "b": {"name": "b", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(2..9)"}, "s1": {"name": "s1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)"}, "ssquares": {"name": "ssquares", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "(a+c)^2+(b+d)^2+(f+g)^2"}, "q": {"name": "q", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "M+N"}}, "parts": [{"prompt": "

Calculate the following.

$\\vert \\mathbf{v} \\vert=$ [[0]]

Let $\\mathbf{z}=\\mathbf{v}+\\mathbf{w}$.

Calculate the unit vector $\\mathbf{\\hat{z}}$ in the direction of $\\mathbf{z}$. Write $\\mathbf{\\hat{z}}$ as a row vector.

$\\mathbf{\\hat{z}}= \\big($ [[0]], [[1]], [[2]] $\\big)$

You must enter your answers exactly, using the function sqrt(x) as necessary.

", "adaptiveMarkingPenalty": 0, "scripts": {}, "marks": 0, "customName": "", "useCustomName": false, "sortAnswers": false, "variableReplacements": [], "unitTests": [], "customMarkingAlgorithm": "", "gaps": [{"adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "valuegenerators": [], "answer": "({(a + c)} / Sqrt({(((a + c) ^ 2) + ((b + d) ^ 2) + ((g + f) ^ 2))}))", "showCorrectAnswer": true, "checkingAccuracy": "0.001", "checkVariableNames": false, "marks": "0.25", "customName": "", "useCustomName": false, "unitTests": [], "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "showPreview": true, "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "vsetRange": [0, 1], "vsetRangePoints": 5}, {"adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "valuegenerators": [], "answer": "({(b + d)} / Sqrt({(((a + c) ^ 2) + ((b + d) ^ 2) + ((g + f) ^ 2))}))", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "checkVariableNames": false, "marks": "0.25", "customName": "", "useCustomName": false, "unitTests": [], "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "showPreview": true, "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "vsetRange": [0, 1], "vsetRangePoints": 5}, {"adaptiveMarkingPenalty": 0, "scripts": {}, "variableReplacements": [], "checkingType": "absdiff", "customMarkingAlgorithm": "", "valuegenerators": [], "answer": "({(g + f)} / Sqrt({(((a + c) ^ 2) + ((b + d) ^ 2) + ((g + f) ^ 2))}))", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "checkVariableNames": false, "marks": "0.25", "customName": "", "useCustomName": false, "unitTests": [], "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "showPreview": true, "failureRate": 1, "type": "jme", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "vsetRange": [0, 1], "vsetRangePoints": 5}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showCorrectAnswer": true}], "ungrouped_variables": ["b4", "q", "s3", "s2", "s1", "s5", "s4", "ssquares", "v1", "v2", "a4", "a", "c", "b", "d", "g", "f", "m", "n", "ssquaresb", "ssquaresa", "v"], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"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/"}]}]}], "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"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/"}]}