// Numbas version: exam_results_page_options
{"name": "Cartesian form of the equation of a plane", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["x_0", "v", "w"], "name": "Vectors"}], "variables": {"w": {"group": "Vectors", "templateType": "anything", "definition": "vector(s3,p1,p3)", "name": "w", "description": ""}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "v": {"group": "Vectors", "templateType": "anything", "definition": "vector(0,s1,p2)", "name": "v", "description": ""}, "p1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s7*random(2..5)", "name": "p1", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s4", 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[{"showFeedbackIcon": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>The vector equation of a plane is</p>\n<p>\\[ \\boldsymbol{x}=\\boldsymbol{x_0}+\\lambda \\boldsymbol{v} + \\mu \\boldsymbol{w} \\]</p>\n<p>If you let&nbsp;$\\boldsymbol{n}=\\boldsymbol{v} \\times \\boldsymbol{w}$, then $\\boldsymbol{x} \\cdot \\boldsymbol{n} = \\boldsymbol{x_0} \\cdot \\boldsymbol{n}$ as $\\boldsymbol{v}\\cdot \\boldsymbol{n}=0$ and $\\boldsymbol{w}\\cdot \\boldsymbol{n}=0$.</p>\n<p>If $\\boldsymbol{x}=(x,\\;y,\\;z)$ is the Cartesian representation of a point $\\boldsymbol{x}$ on the plane, the equation of the plane in Cartesian coordinates is then given by:</p>\n<p>\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}&nbsp;\\cdot \\boldsymbol{n} =\\boldsymbol{x_0} \\cdot \\boldsymbol{n}\\]</p>", "marks": 0, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "unitTests": []}], "prompt": "<p>Enter $ax + by + cz$ with $a$, $b$ and $c$ integers, not decimals.</p>\n<p>Equation of the plane: [[0]]&nbsp;$ = $&nbsp;[[1]]</p>\n<p>You can get help by clicking on <em>Show steps</em>.</p>", "stepsPenalty": 0, "unitTests": [], "scripts": {}, "gaps": [{"answer": "{coeffx}*x+{coeffy}*y + {coeffz}*z ", "showCorrectAnswer": true, "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "std", "showPreview": true, "checkVariableNames": false, "unitTests": [], "notallowed": {"message": "<p>Input numbers as integers and not decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "vsetRange": [0, 1], "vsetRangePoints": 5, "checkingType": "absdiff", "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "expectedVariableNames": [], "variableReplacements": [], "marks": "1.5", "showFeedbackIcon": true}, {"answer": "{con}", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "showPreview": true, "failureRate": 1, "checkVariableNames": false, "unitTests": [], "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "checkingType": "absdiff", "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.5", "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>You are given the vector equation of a plane in $\\mathbb{R}^3$:</p>\n<p>\\[ \\boldsymbol{x} = \\var{x_0} + \\lambda \\var{v} + \\mu \\var{w}, \\quad -\\infty\\lt\\lambda,\\;\\mu \\lt \\infty \\]</p>\n<p>In this question you want to find an equation of this plane in the Cartesian from</p>\n<p>\\[ ax + by + cz = d&nbsp;\\]</p>", "tags": ["Cartesian form of the equation of a plane", "checked2015", "cross product", "equation of a plane", "Parametric form for a plane in three space", "unused", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the Cartesian form $ax+by+cz=d$ of the equation of the plane $\\boldsymbol{r=r_0+\\lambda a+\\mu b}$.</p>\n<p>The solution is not unique. The constant on right hand side could be given to ensure that the left hand side is unique.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The vector equation of the plane is</p>\n<p>\\[ \\boldsymbol{x}=\\boldsymbol{x_0}+\\lambda \\boldsymbol{v} + \\mu \\boldsymbol{w} \\]</p>\n<p>where</p>\n<p>\\begin{align}<br/>\\boldsymbol{x_0} &amp;= \\var{x_0}, &amp; \\boldsymbol{v} &amp;= \\var{v}, &amp; \\boldsymbol{w} &amp;= \\var{w}<br/>\\end{align}</p>\n<p>We have</p>\n<p>\\[ \\boldsymbol{n} = \\boldsymbol{v} \\times \\boldsymbol{w} = \\var{v} \\times \\var{w} = \\var{cross(v,w)} \\]</p>\n<p>If you let&nbsp;$\\boldsymbol{n}=\\boldsymbol{v} \\times \\boldsymbol{w}$, then $\\boldsymbol{x} \\cdot \\boldsymbol{n} = \\boldsymbol{x_0} \\cdot \\boldsymbol{n}$ as $\\boldsymbol{v}\\cdot \\boldsymbol{n}=0$ and $\\boldsymbol{w}\\cdot \\boldsymbol{n}=0$.</p>\n<p>If $\\boldsymbol{x}=(x,\\;y,\\;z)$ is the Cartesian representation of a point $\\boldsymbol{x}$ on the plane, the equation of the plane in Cartesian coordinates is then given by:</p>\n<p>\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}&nbsp;\\cdot \\boldsymbol{n} =\\boldsymbol{x_0} \\cdot \\boldsymbol{n}\\]</p>\n<p>That is,</p>\n<p>\\[ \\simplify[all,!noLeadingMinus]{ {coeffx}*x+{coeffy}*y + {coeffz}*z } = \\var{con} \\]</p>", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}