// Numbas version: finer_feedback_settings
{"name": "Scalar triple product of coplanar vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["w1", "w2", "w3"], "name": "w"}, {"variables": ["s1", "s2", "s3", "s4", "s5", "s6"], "name": "s"}, {"variables": ["x1", "x2", "x3"], "name": "x"}, {"variables": ["y1", "y2", "y3"], "name": "y"}, {"variables": ["z1", "z2"], "name": "z"}], "variables": {"s1": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "x2": {"group": "x", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "x2"}, "y3": {"group": "y", "templateType": "anything", "definition": "if(x1=x3,if(y1=ty3,-ty3,ty3),ty3)", "description": "", "name": "y3"}, "w1": {"group": "w", "templateType": "anything", "definition": "x2*y3-y2*x3", "description": "", "name": "w1"}, "s4": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "y2": {"group": "y", "templateType": "anything", "definition": "if(x1*z2=x2*y1,z2+1,z2)", "description": "", "name": "y2"}, "s2": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "z2": {"group": "z", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "z2"}, "x3": {"group": "x", "templateType": "anything", "definition": "random(-9..9)", "description": "", "name": "x3"}, "lambda": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(x3*(y1*z2-y2*z1)+y3*(x2*z1-x1*z2))/(x2*y1-x1*y2)", "description": "", "name": "lambda"}, "w2": {"group": "w", "templateType": "anything", "definition": "x3*y1-y3*x1", "description": "", "name": "w2"}, "x1": {"group": "x", "templateType": "anything", "definition": "s1*random(1..9)", "description": "", "name": "x1"}, "s5": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s5"}, "s3": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "s6": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s6"}, "y1": {"group": "y", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "y1"}, "z1": {"group": "z", "templateType": "anything", "definition": "s5*random(1..9)", "description": "", "name": "z1"}, "w3": {"group": "w", "templateType": "anything", "definition": "x1*y2-x2*y1", "description": "", "name": "w3"}, "ty3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s6*random(1..9)", "description": "", "name": "ty3"}}, "ungrouped_variables": ["ty3", "lambda"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Scalar triple product of coplanar vectors", "showQuestionGroupNames": false, "functions": {}, "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "allowFractions": true, "type": "numberentry", "showCorrectAnswer": true, "minValue": "lambda", "correctAnswerFraction": true, "variableReplacements": [], "marks": 2, "maxValue": "lambda"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>$\\lambda=$&nbsp;[[0]].</p>\n<p>Enter&nbsp;your answer as a&nbsp;fraction or integer and not a decimal.</p>", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are given three points in $\\mathbb{R}^3$,</p>\n<p>\\begin{align}<br/>\\boldsymbol{a} &amp;= \\var{vector(x1,x2,x3)}, &amp;<br/>\\boldsymbol{b} &amp;= \\var{vector(y1,y2,y3)}, &amp;<br/>\\boldsymbol{c} &amp;= \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix}<br/>\\end{align}</p>\n<p>where $\\lambda$ is a parameter to be determined.</p>\n<p>Find the&nbsp;value of $\\lambda$ such that $\\boldsymbol{a}$, $\\boldsymbol{b}$ and $\\boldsymbol{c}$ all lie on the same plane through the origin.</p>", "tags": ["checked2015", "colinear", "colinearvectors", "cross product", "determining if three vectors in three space are colinear", "dot product", "inner product", "scalar product", "scalar triple product", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>14/7/2015</strong></p>\n<p>Added module tag</p>\n<p><strong></strong></p>\n<p><strong>03/12/2013</strong></p>\n<p>Clarified wording vector -&gt; position vector. (AJY)</p>\n<p>Typo colinear -&gt; collinear. (AJY)</p>\n<p><strong>15/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>Corrected error. It was possible for the third coordinate of $A \\times B$ &nbsp;to be 0. Hence could not be colinear in some circumstances.&nbsp;Calculations checked as a result.</p>\n<p><strong>16/07/2012:</strong></p>\n<p>Added tags.<strong><br /></strong></p>\n<p>Moved \\rightarrow so that it is located at the beginning of the line.</p>\n<p>Question appears to be working correctly.</p>\n<p><strong>27/08/2012:</strong></p>\n<p>Changed coplanar to colinear - for obvious reasons!!</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Three 3 dim vectors, one with a parameter $\\lambda$ in the third coordinate. Find value of $\\lambda$ ensuring vectors coplanar. Scalar triple product.</p>"}, "advice": "<p>Note that $\\boldsymbol{a}\\times \\boldsymbol{b}$ is a vector which is perpendicular to both $\\boldsymbol{a}$ and $\\boldsymbol{b}$ and hence to the plane through the origin&nbsp;containing $\\boldsymbol{a}$ and $\\boldsymbol{b}$.&nbsp;</p>\n<p>So if $\\boldsymbol{c}$ is perpendicular to $\\boldsymbol{a} \\times \\boldsymbol{b}$, i.e. $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} = 0$, it must lie on the same plane.</p>\n<p>Now</p>\n<p>\\begin{align}&nbsp;<br/>\\boldsymbol{a} \\times \\boldsymbol{b} &amp;= \\var{vector(x1,x2,x3)} \\times \\var{vector(y1,y2,y3)} \\\\[1em]<br/> &amp;= \\simplify[]{vector({x2}*{y3}-{x3}*{y2}, {x3}*{y1}-{x1}*{y3}, {x1}*{y2}-{x2}*{y1})} \\\\[1em]<br/>&amp;= \\var{vector(w1,w2,w3)}<br/>\\end{align}</p>\n<p>Hence</p>\n<p>\\begin{align}<br/>(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &amp;= \\var{vector(w1,w2,w3)} \\boldsymbol{\\cdot} \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix} \\\\[1em]<br/>&amp;= \\simplify[]{{w1}*{z1}+{w2}*{z2}+{w3}*lambda} \\\\[1em]<br/>&amp;= \\simplify{{w1*z1+w2*z2}+{w3}*lambda}<br/>\\end{align}</p>\n<p><br/>We now require a value of $\\lambda$ so that $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c}=0$.</p>\n<p>\\begin{align}<br/>&amp;&amp;(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &amp;= &nbsp;0 \\\\<br/>\\implies &amp;&amp;\\simplify{{w1*z1+w2*z2}+{w3}*lambda} &amp;= 0 \\\\<br/> \\implies &amp;&amp;\\lambda &amp;= \\simplify[std]{{-w1*z1-w2*z2}/{w3}}<br/>\\end{align}</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}