// Numbas version: finer_feedback_settings
{"name": "Harry's copy of 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": [{"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>", "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [], "name": ""}], "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>", "type": "question", "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "lambda", "mu1", "mu2", "v", "w", "u"], "functions": {}, "variable_groups": [], "tags": ["checked2015", "dot product", "finding perpendicular vectors", "inner product", "mas1602", "MAS1602", "perpendicular vectors", "product", "scalar product", "vectors"], "variablesTest": {"maxRuns": 100, "condition": "u<>vector(0,0,0)"}, "name": "Harry's copy of Determine if vectors are perpendicular", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"marks": 0, "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>", "gaps": [{"marks": 1.5, "variableReplacements": [], "correctAnswerFraction": true, "showPrecisionHint": false, "allowFractions": true, "minValue": "lambda", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "scripts": {}, "maxValue": "lambda", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true}, {"marks": 0, "type": "gapfill", "prompt": "<p>Find $\\lambda \\in \\mathbb{R}$ such that the vector $\\boldsymbol{u} = \\simplify[fractionnumbers]{{u}}$ is contained in the&nbsp;plane through the origin parallel to $\\boldsymbol{v}$ and $\\boldsymbol{w}$.</p>\n<p>$\\lambda =$ [[0]]</p>", "gaps": [{"marks": 0.5, "variableReplacements": [], "correctAnswerFraction": true, "showPrecisionHint": false, "allowFractions": true, "minValue": "lambda", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "scripts": {}, "maxValue": "lambda", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true}], "variables": {"s5": {"definition": "random(1,-1)", "name": "s5", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "mu1": {"definition": "lcm(random(-5..5 except 0),f)", "name": "mu1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "w": {"definition": "vector(c,d,f)", "name": "w", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "g": {"definition": "s1*random(2..9)", "name": "g", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s4": {"definition": "random(1,-1)", "name": "s4", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "f": {"definition": "random(2,4,5,10)", "name": "f", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s1": {"definition": "random(1,-1)", "name": "s1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c": {"definition": "s3*random(2..9)", "name": "c", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "mu2": {"definition": "lcm(random(-5..5 except 0),f)", "name": "mu2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b": {"definition": "s2*random(2..9)", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s3": {"definition": "random(1,-1)", "name": "s3", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "v": {"definition": "vector(a,b,lambda)", "name": "v", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "u": {"definition": "mu1*v+mu2*w", "name": "u", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s2": {"definition": "random(1,-1)", "name": "s2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a": {"definition": "s1*random(2..9)", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "d": {"definition": "s4*random(2..9)", "name": "d", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "lambda": {"definition": "(-a*c-b*d)/f", "name": "lambda", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "preamble": {"css": "", "js": ""}, "showQuestionGroupNames": false, "metadata": {"notes": "<p><strong>14/7/2015</strong></p>\n<p>Added module tag</p>\n<p><strong></strong></p>\n<p><strong>15/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Last part is too easy.</p>\n<p><strong>16/07/2012:</strong></p>\n<p><strong>Added tags.</strong></p>\n<p><strong>Question appears to be working correctly.<br /></strong></p>\n<p><strong>Agree that last part is too easy.<br /></strong></p>", "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>", "licence": "Creative Commons Attribution 4.0 International"}, "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}