// Numbas version: finer_feedback_settings
{"name": "Find a unit normal vector to a surface given in Cartesian form", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"n2hat": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sign(n2[0])*vector(precround(n2[0]/lenn2,4),precround(n2[1]/lenn2,4),precround(n2[2]/lenn2,4))", "description": "", "name": "n2hat"}, "p4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "p4"}, "p13": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "name": "p13"}, "p1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "name": "p1"}, "p12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "p12"}, "p9": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p9"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p3"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "n2andmultiply": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(n2[0]<0,\", and multiply by $-1$,\",\",\")", "description": "", "name": "n2andmultiply"}, "n1hat": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sign(n1[0])*vector(precround(n1[0]/lenn1,3),precround(n1[1]/lenn1,3),precround(n1[2]/lenn1,3))", "description": "", "name": "n1hat"}, "n1andmultiply": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(n1[0]<0,\", and multiply by $-1$,\",\",\")", "description": "", "name": "n1andmultiply"}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(-p10*a2*q[0]^(p10-1)*q[1]^p11-p12*b2*q[0]^(p12-1)*q[1]^p13,-p11*a2*q[0]^p10*q[1]^(p11-1)-p13*b2*q[0]^p12*q[1]^(p13-1),1)", "description": "", "name": "n2"}, "p11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "name": "p11"}, "q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(random(1..9)*sign(random(-1,1)),random(1..9)*sign(random(-1,1)),random(1..9)*sign(random(-1,1)))", "description": "", "name": "q"}, "lenn2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(n2)", "description": "", "name": "lenn2"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a2"}, "p8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p8"}, "p7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "p7"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except a2)*sign(random(-1,1))", "description": "", "name": "b2"}, "p5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p5"}, "p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p2"}, "lenn1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(n1)", "description": "", "name": "lenn1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(random(1..9)*sign(random(-1,1)),random(1..9)*sign(random(-1,1)),random(1..9)*sign(random(-1,1)))", "description": "", "name": "r"}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(p1*a1*r[0]^(p1-1)*r[1]^p2*r[2]^p3+p4*b1*r[0]^(p4-1)*r[1]^p5*r[2]^p6+p7*c1*r[0]^(p7-1)*r[1]^p8*r[2]^p9,p2*a1*r[0]^p1*r[1]^(p2-1)*r[2]^p3+p5*b1*r[0]^p4*r[1]^(p5-1)*r[2]^p6+p8*c1*r[0]^p7*r[1]^(p8-1)*r[2]^p9,p3*a1*r[0]^p1*r[1]^p2*r[2]^(p3-1)+p6*b1*r[0]^p4*r[1]^p5*r[2]^(p6-1)+p9*c1*r[0]^p7*r[1]^p8*r[2]^(p9-1))", "description": "", "name": "n1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "p6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "name": "p6"}, "p10": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "name": "p10"}}, "ungrouped_variables": ["n1hat", "n2hat", "n1andmultiply", "b1", "b2", "lenn2", "lenn1", "r", "p10", "a1", "p11", "p12", "p13", "c1", "p2", "p3", "p1", "p6", "p7", "p4", "p5", "p8", "p9", "q", "a2", "n1", "n2", "n2andmultiply"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find a unit normal vector to a surface given in Cartesian form", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n1hat[0]+0.001", "minValue": "n1hat[0]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n1hat[1]+0.001", "minValue": "n1hat[1]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n1hat[2]+0.001", "minValue": "n1hat[2]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$\\simplify{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}+{c1}*x^{p7}*y^{p8}*z^{p9}}=0$ at $\\pmatrix{\\var{r[0]},\\var{r[1]},\\var{r[2]}}$.</p>\n        <p>$\\boldsymbol{\\hat{n}}=($[[0]]$,$[[1]]$,$[[2]]$)$. &nbsp;(Enter your answers to 3d.p.)</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n2hat[0]+0.0001", "minValue": "n2hat[0]-0.0001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n2hat[1]+0.0001", "minValue": "n2hat[1]-0.0001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "n2hat[2]+0.0001", "minValue": "n2hat[2]-0.0001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$z=\\simplify{{a2}*x^{p10}*y^{p11}+{b2}*x^{p12}*y^{p13}}$ at $\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$.</p>\n        <p>$\\boldsymbol{\\hat{n}}=($[[0]]$,$[[1]]$,$[[2]]$)$. &nbsp;(Enter your answers to 4d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find a unit normal vector $\\boldsymbol{\\hat{n}}$,&nbsp;<strong>with positive $x$-component</strong>, to the following surfaces, at the given point.</p>", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>19/11/2013</strong></p>\n    <p>Set p1/=0, p4=p7=0, so that the x-derivative in part a) is never zero. &nbsp;This avoids division by zero when calculating n1hat.</p>\n    <p>q[1]-&gt;q[0] in definition of n2.</p>\n    <p>Set p12=0, so that the x-derivative in part b) is never zero. &nbsp;This avoids division by zero when calculating n2hat.</p>\n    <p><strong>12/11/2013</strong></p>\n    <p>Set a2 not equal to b2, to avoid any degenerate surfaces in part b).</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Unit normal vector to a surface, given in Cartesian form.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p><strong>Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.</strong></p>\n    <p>a)</p>\n    <p>The surface is given in the form $f(x,y,z)=0$, and so a normal vector to this surface is given by</p>\n    <p>\\[\\begin{align}\\boldsymbol{n}=\\boldsymbol{\\nabla}f&amp;=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}\\\\&amp;=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{p7*c1}*x^{p7-1}*y^{p8}*z^{p9}},\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{p8*c1}*x^{p7}*y^{p8-1}*z^{p9}},\\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{p9*c1}*x^{p7}*y^{p8}*z^{p9-1}}}.\\end{align}\\]</p>\n    <p>Then</p>\n    <p>\\[\\boldsymbol{n}\\vert_{\\pmatrix{\\var{r[0]},\\var{r[1]},\\var{r[2]}}}=\\pmatrix{\\var{n1[0]},\\var{n1[1]},\\var{n1[2]}}\\]</p>\n    <p>is one such normal to the surface at the given point, by straight forward substitution of the components of the point into the components of the normal.</p>\n    <p>The unit normal vector to the surface at the given point is given by $\\boldsymbol{\\hat{n}}=\\frac{\\boldsymbol{n}}{\\lvert\\boldsymbol{n}\\rvert}$, where</p>\n    <p>\\[\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}.\\]</p>\n    <p>In this case</p>\n    <p>\\[\\lvert\\boldsymbol{n}\\rvert=\\sqrt{(\\var{n1[0]})^2+(\\var{n1[1]})^2+(\\var{n1[2]})^2}=\\var{precround(lenn1,3)}\\;\\text{to 3d.p.}\\]</p>\n    <p>Now divide each component of $\\boldsymbol{n}$ by this value{n1andmultiply} so that the normal vector to the surface at the given point, <strong>with positive $x$-component</strong> is</p>\n    <p>\\[\\boldsymbol{\\hat{n}}=\\pmatrix{\\var{n1hat[0]},\\var{n1hat[1]},\\var{n1hat[2]}}\\;\\text{to 3d.p.}\\]</p>\n    <p>&nbsp;</p>\n    <p>b)</p>\n    <p>The surface is given in the form $z=g(x,y)$. &nbsp;This expression can either be rearranged into the form $f(x,y,z)=0$, and then adopt the same method of solution as in part a) or, alternatively, a normal vector to this surface is given by</p>\n    <p>\\[\\boldsymbol{n}=\\pmatrix{-\\frac{\\partial g}{\\partial x},-\\frac{\\partial g}{\\partial y},1}=\\pmatrix{\\simplify{{-p10*a2}*x^{p10-1}*y^{p11}-{p12*b2}*x^{p12-1}*y^{p13}},\\simplify{{-p11*a2}*x^{p10}*y^{p11-1}-{p13*b2}*x^{p12}*y^{p13-1}},1}.\\]</p>\n    <p>Then</p>\n    <p>\\[\\boldsymbol{n}\\vert_{\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}}=\\pmatrix{\\var{n2[0]},\\var{n2[1]},\\var{n2[2]}}\\]</p>\n    <p>is one such normal to the surface at the given point, by straight forward substitution of the components of the point into the components of the normal.</p>\n    <p>The unit normal vector to the surface at the given point is given by $\\boldsymbol{\\hat{n}}=\\frac{\\boldsymbol{n}}{\\lvert\\boldsymbol{n}\\rvert}$, where</p>\n    <p>\\[\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}.\\]</p>\n    <p>In this case</p>\n    <p>\\[\\lvert\\boldsymbol{n}\\rvert=\\sqrt{(\\var{n2[0]})^2+(\\var{n2[1]})^2+(\\var{n2[2]})^2}=\\var{precround(lenn2,4)}\\;\\text{to 4d.p.}\\]</p>\n    <p>Now divide each component of $\\boldsymbol{n}$ by this value{n2andmultiply} so that the normal vector to the surface at the given point,&nbsp;<strong>with positive $x$-component</strong>&nbsp;is</p>\n    <p>\\[\\boldsymbol{\\hat{n}}=\\pmatrix{\\var{n2hat[0]},\\var{n2hat[1]},\\var{n2hat[2]}}\\;\\text{to 4d.p.}.\\]</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/"}]}