// Numbas version: exam_results_page_options
{"name": "Find the angle between planes", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$\\alpha = $ [[0]]&nbsp;radians</p>\n<p><em>(Enter your answer in radians, to 3 decimal places)</em></p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "mustBeReduced": false, "type": "numberentry", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "allowFractions": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "minValue": "ans", "maxValue": "ans", "precision": 3, "unitTests": [], "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "marks": 2, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.001", "description": "", "name": "tol"}, "r1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(-1,1)*random(1..9),3))", "description": "", "name": "r1"}, "n1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(r2-r1,r3-r1)", "description": "", "name": "n1"}, "tn1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(r2-r1,tr3-r1)", "description": "", "name": "tn1"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "b1"}, "tr3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(sign(r1[0])*random(1..9),sign(r1[1])*random(1..5),sign(r1[2])*random(1..9))", "description": "", "name": "tr3"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "a1"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "d1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)*random(1..9)", "description": "", "name": "c1"}, "r3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if (tn1=vector(0), vector(tr3[0]+1,tr3[1],tr3[2]), tr3)", "description": "", "name": "r3"}, "ans": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(alpha>pi/2,pi-alpha,alpha)", "description": "", "name": "ans"}, "r2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(sign(r1[0])*random(1..9 except abs(r1[0])),sign(r1[1])*random(2..5),sign(r1[2])*random(1..5))", "description": "", "name": "r2"}, "alpha": {"group": "Ungrouped variables", "templateType": "anything", "definition": "arccos((a1*n1[0]+b1*n1[1]+c1*n1[2])/sqrt((abs(n1)^2)*(a1^2+b1^2+c1^2)))", "description": "", "name": "alpha"}}, "ungrouped_variables": ["tn1", "r1", "r2", "r3", "a1", "b1", "tol", "alpha", "ans", "n1", "c1", "tr3", "d1"], "name": "Find the angle between planes", "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Calculate the angle $\\alpha,\\;0\\leq\\alpha \\leq \\frac{\\pi}{2}$, between the plane $\\Pi_1$, passing through the points</p>\n<p>\\begin{align}<br/>\\boldsymbol{r_1} &amp;= \\var{r1}, &amp; \\boldsymbol{r_2} &amp;= \\var{r2}, &amp;&nbsp;\\boldsymbol{r_3} &amp;= \\var{r3}<br/>\\end{align}</p>\n<p>and the plane, $\\Pi_2$, whose equation is</p>\n<p>\\[\\simplify[std]{{a1}x+{b1}y+{c1}z={d1}}\\]</p>", "tags": ["angle between lines with a common point in 3 space", "angle between planes", "cartesian equation of a plane", "checked2015", "diagram needed", "finding the angle between two planes", "normal to a plane", "parametric form of a plane", "plane given by three points in three space", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find angle between plane $\\Pi_1$, given by three points, and the plane $\\Pi_2$ given in Cartesian form.</p>\n<p>The calculation of $cos(\\alpha)$ at the end of Advice has fractionNumbers switched on and so the result is presented as a fraction, which can be misleading. Best if calculation is followed through without using fractionNumbers.</p>"}, "advice": "<p>The angle between two planes is given by the angle between their normals.</p>\n<p>The plane $\\Pi_1$ can be written 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>and the normal $\\boldsymbol{n}_1$ to this plane is given by:</p>\n<p>\\[ \\boldsymbol{n}_1 = (\\boldsymbol{r}_2 - \\boldsymbol{r}_1)\\times (\\boldsymbol{r}_3- \\boldsymbol{r}_1)\\]</p>\n<p>For $ \\boldsymbol{r}_1$, $\\boldsymbol{r}_2$, $\\boldsymbol{r}_3$ as given.</p>\n<p>\\[ \\boldsymbol{n}_1 = \\var{n1}&nbsp;\\]</p>\n<p>The normal to the plane $\\Pi_2$ is given by</p>\n<p>\\[ \\boldsymbol{n}_2 = \\var{vector(a1,b1,c1)} \\]</p>\n<p>The angle between the two normals (and hence the two planes) can be found using:</p>\n<p>\\[ \\cos(\\alpha) = \\frac{\\boldsymbol{n}_1 \\cdot \\boldsymbol{n}_2}{\\lVert\\boldsymbol{n}_1\\rVert \\lVert\\boldsymbol{n}_2\\rVert} \\]</p>\n<p>On calculating this, we&nbsp;obtain</p>\n<p>\\begin{align}<br/>\\boldsymbol{n}_1 \\boldsymbol{\\cdot} \\boldsymbol{n}_2 &amp;= \\var{a1*n1[0]+b1*n1[1]+c1*n1[2]} \\\\<br/>\\lVert\\boldsymbol{n}_1\\rVert &amp;= \\simplify[std]{sqrt({n1[0]^2+n1[1]^2+n1[2]^2})} \\\\<br/> \\lVert\\boldsymbol{n}_2\\rVert &amp;= \\simplify[std]{sqrt({a1^2+b1^2+c1^2})}\\\\<br/>\\cos(\\alpha) &amp;= \\simplify[std]{{(a1*n1[0]+b1*n1[1]+c1*n1[2])/sqrt((n1[0]^2+n1[1]^2+n1[2]^2)*(a1^2+b1^2+c1^2))}}<br/>\\end{align}</p>\n<p>Now calculate $\\arccos(\\alpha) = \\var{precround(alpha,3)}$. The angle returned by your calculator will give a value between $0$ and $\\pi$. If it's bigger than $\\frac{\\pi}{2}$, subtract the&nbsp;calculated value from $\\pi$ to obtain an acute angle. So the angle between the two planes is&nbsp;</p>\n<p><span data-jme-visible=\"alpha&gt;pi/2\">\\[ \\alpha' = \\pi - \\var{precround(alpha,3)} = \\var{precround(ans,3)} \\text{ radians} \\]</span></p>\n<p><span data-jme-visible=\"alpha&lt;=pi/2\">\\[ \\alpha = \\var{precround(ans,3)} \\text{ radians} \\]</span></p>\n<p>to 3 decimal places.</p>\n<p></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/"}]}