// Numbas version: exam_results_page_options
{"name": "Vector Algebra - Practice-2.1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Vector Algebra - Practice-2.1", "tags": [], "metadata": {"description": "Basic rules of vector algebra", "licence": "None specified"}, "statement": "<p>Let $\\var{vi},\\var{vj},\\var{vk}$ form an orthonormal basis. Consider<br/>\\[<br/>&nbsp; &nbsp; \\var{va}=\\var{disp(a)}, \\qquad \\var{vb}= \\var{disp(b)} , \\qquad \\var{vc}=\\var{{ccoef}*a[0]}\\var{vi} -\\var{lcoef} t \\var{vj}-\\var{abs({ccoef}*a[2])}\\var{vk},<br/>\\]<br/>where $t\\in\\mathbb{R}$.</p>", "advice": "<p>a) Let $\\var{vah}$ be the unit vector corresponding to $\\var{va}$, i.e. <br/>\\[<br/>\\var{vah}=\\frac{1}{|\\var{va}|}\\var{va}.<br/>\\]<br/>Since $\\var{vd}$ is opposite to $\\var{va}$, it is also opposite to $\\var{vah}$. Since $|\\var{vd}|=\\simplify{{acoef}*{alen}}$, we conclude that<br/>\\[<br/>\\var{vd} = -\\simplify{{acoef}*{alen}}\\var{vah}=-\\frac{\\simplify{{acoef}*{alen}}}{|\\var{va}|}\\var{va}.<br/>\\]<br/>We have:<br/>\\[<br/>|\\var{va}|=\\sqrt{\\var{abs(a[0])}^2+\\var{abs(a[1])}^2+\\var{abs(a[2])}^2}=\\var{alen}.<br/>\\]<br/>Therefore,<br/>\\[<br/>\\var{vd} =-\\frac{\\simplify{{acoef}*{alen}}}{\\var{alen}}\\bigl(\\var{disp(a)}\\bigr)=\\var{disp(apar)}.<br/>\\]</p>\n<p></p>\n<p style=\"text-align: left;\">b) We know that<br/>\\[<br/>\\cos \\theta =\\frac{\\var{va}\\cdot\\var{vb}}{|\\var{va}|\\, |\\var{vb}|}.<br/>\\]<br/>We have found that $|\\var{va}|=\\var{alen}$. Next,<br/>\\[<br/>|\\var{vb}|=\\sqrt{\\var{abs(b[0])}^2+\\var{abs(b[1])}^2+\\var{abs(b[2])}^2}=\\var{blen}.<br/>\\]<br/>We have also that<br/>\\[<br/>\\var{va}\\cdot\\var{vb}=\\bigl(\\var{disp(a)}\\bigr)\\cdot\\bigl(\\var{disp(b)}\\bigr)=\\var{dot(a,b)}<br/>\\]<br/>Therefore,<br/>\\[<br/>\\cos \\theta=\\frac{\\var{dot(a,b)}}{\\var{alen} \\cdot\\var{blen}}=\\var{sfrac(costheta[0],costheta[1])}.<br/>\\]</p>\n<p>c) Since vectors $\\var{va}=\\var{disp(a)}$ and $\\var{vc}=\\var{{ccoef}*a[0]}\\var{vi} -\\var{lcoef} t \\var{vj}+\\var{abs({ccoef}*a[2])}\\var{vk}$ are parallel, there exists a number $r\\in\\mathrm{R}$ such that<br/>\\[<br/>\\var{vc}=r\\var{va}.<br/>\\]<br/>Equating coefficients before $\\var{vi},\\var{vj},\\var{vk}$, we get<br/>\\[<br/>\\begin{cases} <br/>\\displaystyle \\var{{ccoef}*a[0]} = \\simplify{{a[0]}*r},\\\\[2mm]<br/>\\displaystyle -\\var{lcoef} t =\\simplify{{a[1]}*r},\\\\[2mm]<br/>\\displaystyle \\var{abs({ccoef}*a[2])} =\\simplify{{a[2]}*r},<br/>\\end{cases}<br/>\\qquad\\qquad <br/>\\begin{cases}<br/>\\displaystyle r=\\var{ccoef},\\\\[3mm]<br/> \\displaystyle t = -\\frac{ \\simplify{{a[1]}*r}}{\\var{lcoef}},<br/>\\end{cases}<br/>\\qquad\\qquad t=\\var{sfrac(l[0],l[1])}.<br/>\\]</p>\n<p style=\"text-align: left;\">d) Vectors $\\var{vb}= \\var{disp(b)}$ and $ \\var{vc}=\\var{{ccoef}*a[0]}\\var{vi} -\\var{lcoef} t \\var{vj}+\\var{abs({ccoef}*a[2])}\\var{vk}$ are orthogonal if and only if <br/>\\[<br/>\\var{vb}\\cdot\\var{vc}=0.<br/>\\]<br/>Since<br/>\\[<br/>\\var{vb}\\cdot\\var{vc}=\\bigl( \\var{disp(b)}\\bigr) \\cdot\\bigl( \\var{{ccoef}*a[0]}\\var{vi} -\\var{lcoef} t \\var{vj}+\\var{abs({ccoef}*a[2])}\\var{vk}\\bigr) =\\simplify{{dop} - { dop1} * t},<br/>\\]<br/>we require<br/>\\[<br/>\\simplify{{dop} - { dop1} * t}=0, \\qquad\\qquad&nbsp; t=\\var{sfrac(lort[0],lort[1])}.<br/>\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], 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"can_override": false}, "aset": {"name": "aset", "group": "AGroup", "definition": "shuffle(random([1,2,2],[1,4,8],[2,3,6],[2,4,4],[4,4,7]))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a<>b and abs(dop)<=100 and abs(dop1)<=100", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Vectors", "variables": ["va", "vb", "vc", "vd", "ve", "vi", "vj", "vk", "vn", "vr", "vv", "vah"]}, {"name": "AGroup", "variables": ["a", "acoef", "alen", "apar", "aset", "asign", "b", "blen", "bset", "bsign", "ccoef", "costheta", "l", "lcoef", "lort", "mark", "dop", "dop1"]}], "functions": {"vvec": {"parameters": [["sarg", "string"]], "type": "string", "language": "jme", "definition": "latex('\\\\simplify\\{ underline:boldsymbol:{sarg} \\}')"}, "disp": {"parameters": [["varg", "vector"]], "type": "string", "language": "jme", "definition": "latex(\"\\\\simplify[!noLeadingMinus,unitFactor,zeroFactor,zeroTerm]\\{ {varg[0]} underline:boldsymbol:i + {varg[1]} underline:boldsymbol:j + {varg[2]} underline:boldsymbol:k\\}\")"}, "sfrac": {"parameters": [["nom", "number"], ["denom", "number"]], "type": "string", "language": "jme", "definition": "latex('\\\\simplify[simplifyFractions,unitDenominator]\\{{nom}/{denom}\\}')"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the vector $\\var{vd}$ whose magnitude is $\\simplify{{acoef}*{alen}}$ and which is opposite to $\\var{va}$:</p>\n<p style=\"text-align: center;\">$ \\var{vd}=$[[0]]$\\var{vi}+$[[1]]$\\var{vj}+$[[2]]$\\var{vk}$.</p>\n<p style=\"text-align: center;\"></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{apar[0]}", "maxValue": "{apar[0]}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{apar[1]}", "maxValue": "{apar[1]}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{apar[2]}", "maxValue": "{apar[2]}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let $\\theta$ be the angle between vectors $\\var{va}$ and $\\var{vb}$. Find</p>\n<p style=\"text-align: center;\">$\\cos\\theta = $[[0]]</p>\n<p style=\"text-align: left;\">(To type a fraction use the form 'p/q' or '-p/q' where 'p' and 'q' are natural numbers.)</p>\n<p style=\"text-align: left;\"></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "costheta[0]/costheta[1]", "maxValue": "costheta[0]/costheta[1]", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let $t\\in\\mathbb{R}$ be such that vectors $\\var{va}$ and $\\var{vc}$ are parallel. Find</p>\n<p style=\"text-align: center;\">$t=$[[0]]</p>\n<p style=\"text-align: left;\">(To type a fraction use the form 'p/q' or '-p/q' where 'p' and 'q' are natural numbers.)</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "l[0]/l[1]", "maxValue": "l[0]/l[1]", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let $t\\in\\mathbb{R}$ be such that vectors $\\var{vb}$ and $\\var{vc}$ are orthogonal. Find</p>\n<p style=\"text-align: center;\">$t=$[[0]]</p>\n<p style=\"text-align: left;\">(To type a fraction use the form 'p/q' or '-p/q' where 'p' and 'q' are natural numbers.)</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "lort[0]/lort[1]", "maxValue": "lort[0]/lort[1]", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Dmitri Finkelshtein", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2756/"}, {"name": "Ian Davies", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3119/"}]}]}], "contributors": [{"name": "Dmitri Finkelshtein", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2756/"}, {"name": "Ian Davies", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3119/"}]}