// Numbas version: exam_results_page_options
{"name": "Cartesian equation of a plane", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "extendBaseMarkingAlgorithm": true, "prompt": "<p>We can write a&nbsp;vector equation of the plane in the form:</p>\n<p>$\\boldsymbol{r}=\\boldsymbol{r_1}+\\lambda (\\boldsymbol{r_2}-\\boldsymbol{r_1}) + \\mu (\\boldsymbol{r_3}-\\boldsymbol{r_1})$</p>\n<p>Note that three points determine a plane and</p>\n<ol>\n<li>$\\lambda=0,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_1}$</li>\n<li>$\\lambda=1,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_2}$</li>\n<li>$\\lambda=0,\\;\\;\\mu=1$ gives $\\boldsymbol{r}=\\boldsymbol{r_3}$</li>\n</ol>\n<p>Note that if we let</p>\n<p>\\[\\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1})\\]</p>\n<p>then $\\boldsymbol{n}\\cdot (\\boldsymbol{r_2}-\\boldsymbol{r_1})=0$ and $\\boldsymbol{n}\\cdot (\\boldsymbol{r_3}-\\boldsymbol{r_1})=0$.</p>\n<p>If $\\boldsymbol{r} = (x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that</p>\n<p>\\[ \\boldsymbol{r}\\cdot \\boldsymbol{n}=(x,\\;y,\\;z)\\cdot &nbsp;\\boldsymbol{n}=\\boldsymbol{r_1}\\cdot \\boldsymbol{n} \\]</p>", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true}], "prompt": "<p>Find the Cartesian equation of this plane, in the form $ax+by+cz = d$, 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>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "{coeffx}*x+{coeffy}*y + {coeffz}*z ", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "answerSimplification": "std", "expectedVariableNames": [], "showPreview": true, "notallowed": {"message": "<p>Input numbers as integers and not decimals</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "showFeedbackIcon": true, "scripts": {}, "vsetRangePoints": 5, "type": "jme", "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 2}, {"answer": "{con}", "showCorrectAnswer": true, "failureRate": 1, "customMarkingAlgorithm": "", "checkingType": "absdiff", "vsetRangePoints": 5, "expectedVariableNames": [], "showPreview": true, "checkVariableNames": false, "unitTests": [], "vsetRange": [0, 1], "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "jme", "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"coeffy": {"templateType": "anything", "group": "Answer", "definition": "n[1]", "name": "coeffy", "description": ""}, "b3": {"templateType": "anything", "group": "Vector r_2", "definition": "if(t=2,1,2)", "name": "b3", "description": ""}, "a": {"templateType": "anything", "group": "Vector r_1", "definition": "random(-9..9 except -1..1)", "name": "a", "description": ""}, "s6'": {"templateType": "anything", "group": "Vector r_2", "definition": "random(1,-1)", "name": "s6'", "description": ""}, "t": {"templateType": "anything", "group": "Vector r_1", "definition": "random(0..2)", "name": "t", "description": ""}, "directions": {"templateType": "anything", "group": "Vector r_2", "definition": "map(id(3)[x],x,shuffle(0..2))", "name": "directions", "description": ""}, "r_3": {"templateType": "anything", "group": "Vector r_3", "definition": "vector(random(-9..9 except -1..1),random(-9..9 except -1..1),random(2..9))", "name": "r_3", "description": ""}, "r_2": {"templateType": "anything", "group": "Vector r_2", "definition": "s6*directions[a3] + p1*directions[b3]", "name": "r_2", "description": ""}, "n": {"templateType": "anything", "group": "Answer", "definition": "cross(r_2-r_1,r_3-r_1)", "name": "n", "description": "<p>Normal to the plane</p>"}, "a3": {"templateType": "anything", "group": "Vector r_2", "definition": "if(t=0,1,0)", "name": "a3", "description": ""}, "r_1": {"templateType": "anything", "group": "Vector r_1", "definition": "id(3)[t]*random(-9..9 except -1..1)", "name": "r_1", "description": ""}, "coeffx": {"templateType": "anything", "group": "Answer", "definition": "n[0]", "name": "coeffx", "description": ""}, "coeffz": {"templateType": "anything", "group": "Answer", "definition": "n[2]", "name": "coeffz", "description": ""}, "s6": {"templateType": "anything", "group": "Vector r_2", "definition": "switch(\n  t=1,\n    if(r_3[2]*s6'=r_3[1]*p1,-s6',s6'),\n  t=2,\n    if(r_3[2]*s6'=r_3[0]*p1,-s6',s6'),\n    if(r_3[1]*s6'=r_3[0]*p1,-s6',s6')\n)", "name": "s6", "description": ""}, "p1": {"templateType": "anything", "group": "Vector r_2", "definition": "random(-5..5 except -1..1)", "name": "p1", "description": ""}, "con": {"templateType": "anything", "group": "Answer", "definition": "dot(r_1,n)", "name": "con", "description": ""}}, "ungrouped_variables": [], "name": "Cartesian equation of a plane", "functions": {}, "variable_groups": [{"variables": ["t", "a", "r_1"], "name": "Vector r_1"}, {"variables": ["a3", "b3", "s6'", "s6", "p1", "directions", "r_2"], "name": "Vector r_2"}, {"variables": ["r_3"], "name": "Vector r_3"}, {"variables": ["n", "coeffx", "coeffy", "coeffz", "con"], "name": "Answer"}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>A plane goes through the points:</p>\n<p>\\begin{align}<br/>\\boldsymbol{r_1} &amp;= \\var{r_1}, &amp; \\boldsymbol{r_2} &amp;= \\var{r_2}, &amp; \\boldsymbol{r_3} &amp;= \\var{r_3}<br/>\\end{align}</p>", "tags": ["cartesian equation of a plane", "checked2015", "equation of a plane", "plane through 3 points", "planes", "Steps", "steps", "the equation of a plane", "unused", "Vector", "vector", "vector equation of a plane", "vector geometry", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.</p>\n<p>There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.</p>"}, "extensions": [], "advice": "<p>We can write a&nbsp;vector equation of the plane in the form:</p>\n<p>$\\boldsymbol{r}=\\boldsymbol{r_1}+\\lambda (\\boldsymbol{r_2}-\\boldsymbol{r_1}) + \\mu (\\boldsymbol{r_3}-\\boldsymbol{r_1})$</p>\n<p>Note that three points determine a plane and</p>\n<ol>\n<li>$\\lambda=0,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_1}$</li>\n<li>$\\lambda=1,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_2}$</li>\n<li>$\\lambda=0,\\;\\;\\mu=1$ gives $\\boldsymbol{r}=\\boldsymbol{r_3}$</li>\n</ol>\n<p>Note that if we let</p>\n<p>\\[\\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1})\\]</p>\n<p>then $\\boldsymbol{n}\\cdot (\\boldsymbol{r_2}-\\boldsymbol{r_1})=0$ and $\\boldsymbol{n}\\cdot (\\boldsymbol{r_3}-\\boldsymbol{r_1})=0$.</p>\n<p>Hence $\\boldsymbol{r}\\cdot&nbsp;\\boldsymbol{n}=\\boldsymbol{r_1}\\cdot&nbsp;\\boldsymbol{n}$.</p>\n<p>If $\\boldsymbol{r} = (x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that</p>\n<p>\\[ \\boldsymbol{r}\\cdot \\boldsymbol{n}=(x,\\;y,\\;z)\\cdot &nbsp;\\boldsymbol{n}=\\boldsymbol{r_1}\\cdot \\boldsymbol{n} \\]</p>\n<p>If $\\boldsymbol{r}=(x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that&nbsp;the equation of the plane is given by $\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix}&nbsp;\\cdot \\boldsymbol{n} = \\boldsymbol{r_1} \\cdot \\boldsymbol{n}$.</p>\n<p>We have:</p>\n<p>\\[ \\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1}) = \\var{r_2-r_1} \\times \\var{r_3-r_1} = \\var{n} \\]</p>\n<p>Hence, $\\boldsymbol{r_1} \\cdot \\boldsymbol{n} = \\var{con}$.</p>\n<p>So the Cartesian equation of the plane is</p>\n<p>\\[ \\simplify[all,!noLeadingMinus]{{coeffx}x + {coeffy}y + {coeffz}z = {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/"}]}