// Numbas version: exam_results_page_options
{"name": "Vectors 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\"><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">When are vectors $\\boldsymbol{v,\\;w}$ orthogonal</span></span><span class=\"Apple-style-span\" style=\"-webkit-tap-highlight-color: rgba(26, 26, 26, 0.296875); -webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469);\">?</span></p>"}, "parts": [{"type": "gapfill", "prompt": "<p>Find $\\lambda \\in \\mathbb{R}$ such that $\\boldsymbol{v}$ and $\\boldsymbol{w}$ are orthogonal.</p>\n<p>$\\lambda = $&nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "gaps": [{"notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "minValue": "lambda", "correctAnswerFraction": true, "maxValue": "lambda", "marks": 1.5, "scripts": {}, "allowFractions": true, "mustBeReducedPC": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReduced": false, "showCorrectAnswer": true, "showFeedbackIcon": true}], "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}}], "preamble": {"css": "", "js": ""}, "extensions": [], "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "lambda", "mu1", "mu2", "v", "w", "u"], "statement": "<p>You are given the vectors $\\boldsymbol{v} = \\begin{pmatrix}\\var{a} \\\\ \\var{b} \\\\ \\lambda \\end{pmatrix}$ and $\\boldsymbol{w} = \\begin{pmatrix} \\var{c} \\\\ \\var{d} \\\\ \\var{f} \\end{pmatrix}$.</p>\n<p><em>Enter your answers to the following questions as fractions or integers, not decimals.</em></p>", "variables": {"g": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "g", "definition": "s1*random(2..9)"}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s1", "definition": "random(1,-1)"}, "u": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "u", "definition": "mu1*v+mu2*w"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d", "definition": "s4*random(2..9)"}, "mu2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "mu2", "definition": "lcm(random(-5..5 except 0),f)"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a", "definition": "s1*random(2..9)"}, "mu1": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "mu1", "definition": "lcm(random(-5..5 except 0),f)"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s3", "definition": "random(1,-1)"}, "v": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "v", "definition": "vector(a,b,lambda)"}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s5", "definition": "random(1,-1)"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s4", "definition": "random(1,-1)"}, "w": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "w", "definition": "vector(c,d,f)"}, "lambda": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "lambda", "definition": "(-a*c-b*d)/f"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b", "definition": "s2*random(2..9)"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c", "definition": "s3*random(2..9)"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "f", "definition": "random(2,4,5,10)"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s2", "definition": "random(1,-1)"}}, "advice": "<h4>a)</h4>\n<p>$\\boldsymbol{v}$ and $\\boldsymbol{w}$ are perpendicular to one another when $\\boldsymbol{v} \\cdot \\boldsymbol{w} = 0$.</p>\n<p>Now</p>\n<p>\\begin{align}<br/>\\boldsymbol{v} \\cdot \\boldsymbol{w} &amp;= \\simplify[]{{a}*{c}+{b}*{d}+lambda*{f}} \\\\ <br/>&amp;= \\simplify[std]{{f}*lambda+{a*c+b*d}}<br/>\\end{align}</p>\n<p>Hence</p>\n<p>\\[\\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}}\\]</p>\n<h4>b)</h4>\n<p>$\\boldsymbol{v}$ is in the $xy$ plane when $\\lambda=0$.</p>", "variablesTest": {"maxRuns": 100, "condition": "u<>vector(0,0,0)"}, "name": "Vectors 2", "functions": {}, "variable_groups": [], "type": "question", "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}]}]}], "contributors": [{"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}]}