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{"name": "MATH6005 Assessment 2_Q1of6", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "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": {"condition": "", "maxRuns": 100}, "variable_groups": [], "variables": {"f": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "f", "definition": "random(2..9)"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "g", "definition": "s1*random(2..9)"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s4", "definition": "random(1,-1)"}, "q": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "q", "definition": "M+N"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a", "definition": "s1*random(2..9)"}, "a4": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a4", "definition": "random(3..9)"}, "v2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "v2", "definition": "vector(c,d,f)"}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s1", "definition": "random(1,-1)"}, "ssquaresb": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ssquaresb", "definition": "(c)^2+(d)^2+(f)^2"}, "b4": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b4", "definition": "-random(3..9)"}, "v": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "v", "definition": "v1+v2"}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s5", "definition": "random(1,-1)"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b", "definition": "s2*random(2..9)"}, "ssquaresa": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ssquaresa", "definition": "(a)^2+(b)^2+(g)^2"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s2", "definition": "random(1,-1)"}, "n": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "n", "definition": "matrix([a,b],[c,d])"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s3", "definition": "random(1,-1)"}, "ssquares": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ssquares", "definition": "(a+c)^2+(b+d)^2+(f+g)^2"}, "v1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "v1", "definition": "vector(a,b,g)"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d", "definition": "s4*random(2..9)"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c", "definition": "s3*random(2..9)"}, "m": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "m", "definition": "matrix([a,b],[c,d])"}}, "preamble": {"css": "", "js": ""}, "extensions": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Elementary operations on vectors; sum, modulus, unit vector, scalar multiple.&nbsp;</p>"}, "parts": [{"showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>Calculate&nbsp;$\\mathbf{v}+\\mathbf{w} = $ [[0]]</p>", "marks": 0, "sortAnswers": false, "gaps": [{"showCorrectAnswer": true, "correctAnswer": "v", "marks": "0.25", "numRows": "3", "correctAnswerFractions": false, "type": "matrix", "extendBaseMarkingAlgorithm": true, "scripts": {}, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "unitTests": [], "allowResize": false, "customMarkingAlgorithm": "", "variableReplacements": [], "numColumns": 1, "markPerCell": false, "allowFractions": false}], "type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": []}, {"showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>Calculate&nbsp;$4\\mathbf{v}-2\\mathbf{w} = $[[0]]</p>", "marks": 0, "sortAnswers": false, "gaps": [{"showCorrectAnswer": true, "correctAnswer": "4*v1 - 2*v2", "marks": "0.25", "numRows": "3", "correctAnswerFractions": false, "type": "matrix", "extendBaseMarkingAlgorithm": true, "scripts": {}, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "unitTests": [], "allowResize": false, "customMarkingAlgorithm": "", "variableReplacements": [], "numColumns": 1, "markPerCell": false, "allowFractions": false}], "type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacements": []}], "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>", "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>", "functions": {}, "name": "MATH6005 Assessment 2_Q1of6", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "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/"}]}]}], "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/"}]}