// Numbas version: exam_results_page_options {"name": "Ugur's copy of Find points of intersection, tangents, and angles between parametric curves", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Find points of intersection, tangents, and angles between parametric curves", "tags": ["checked2015", "intersection of curves", "parametric curves", "tangent vectors"], "metadata": {"description": "

Intersection points, tangent vectors, angles between pairs of curves, given in parametric form.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The pair of curves

\\[\\begin{align}\\mathcal{C}_1&:t\\mapsto\\pmatrix{\\simplify{{a1}*t},\\simplify{{b1}*t},\\simplify{{c1}*t}},-\\infty\\leqslant t\\leqslant\\infty\\\\\\mathcal{C}_2&:\\tau\\mapsto\\pmatrix{\\simplify{{d1}*tau},\\simplify{{e1}*tau^2},\\simplify{{f1}*tau^3}},-\\infty\\leqslant \\tau\\leqslant\\infty\\end{align}\\]

intersect at two distinct points $\\boldsymbol{p}$ and $\\boldsymbol{q}$.

", "advice": "

Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.

The two curves $\\mathcal{C}_1$ and $\\mathcal{C}_2$ intersect where

\\[\\begin{align}\\simplify{{a1}*t}&=\\simplify{{d1}*tau}\\tag{1},\\\\\\simplify{{b1}t}&=\\simplify{{e1}*tau^2},\\tag{2}\\\\\\simplify{{c1}*t}&=\\simplify{{f1}*tau^3}.\\tag{3}\\end{align}\\]

From equation (1)

\\[\\tau=\\frac{\\var{a1}}{\\var{d1}}t=\\simplify{{a1}/{d1}t},\\tag{4}\\]

which we substitute into equation (2) to determine that

\\[\\var{b1}t=\\var{e1}\\times\\left(\\simplify{{a1}/{d1}t}\\right)^2=\\simplify{{e1*a1^2}/{d1^2}t^2}.\\]

Then either $t=0$ or $t=\\simplify{{b1*d1^2}/{e1*a1^2}}$.

Substitute these two expressions into equation (4), then either $\\tau=0$ (when $t=0$), or $\\tau=\\simplify{{b1*d1}/{e1*a1}}$ (when $t=\\var{t}$).

(As a check, substitute these pairs of values into equation (3), to show that equality holds.)

To determine the intersection points $\\boldsymbol{p}$ and $\\boldsymbol{q}$, substitute the values of $t$ and $\\tau$ into either expression for the curves $\\mathcal{C}_1$ and $\\mathcal{C}_2$.

The point $\\boldsymbol{p}$ is given by the least value of $t$, which is $t=0$ (and correspondingly $\\tau=0$). The point $\\boldsymbol{p}$ is therefore $\\boldsymbol{p}=\\pmatrix{0,0,0}$.

The point $\\boldsymbol{q}$ is given by the greatest value of $t$, which is $t=\\var{t}$ (and correspondingly $\\tau=\\var{tau}$). The point $\\boldsymbol{q}$ is therefore $\\boldsymbol{q}=\\pmatrix{\\var{a1}\\times\\var{t},\\var{b1}\\times\\var{t},\\var{c1}\\times\\var{t}}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$.

In general, the tangent vector $\\boldsymbol{u}$, of a curve $t\\rightarrow\\pmatrix{x(t),y(t),z(t)}$, is given by $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t},\\frac{\\mathrm{d}z}{\\mathrm{d}t}}$.

The tangent vector of the curve $\\mathcal{C}_1$ is therefore given by $\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$, which is constant, and independent of $t$.

The tangent vector of $\\mathcal{C}_2$ is given by $\\pmatrix{\\var{d1},\\var{2*e1}\\tau,\\var{3*f1}\\tau^2}$, so the tangent vector at the point $\\boldsymbol{p}$ (where $\\tau=0$) is given by $\\boldsymbol{v}=\\pmatrix{\\var{v[0]},\\var{v[1]},\\var{v[2]}}$.

In a similar way, the tangent vector of $\\mathcal{C}_2$ at the point $\\boldsymbol{q}$ (where $\\tau=\\var{tau}$) is given by $\\boldsymbol{w}=\\pmatrix{\\var{w[0]},\\var{w[1]},\\var{w[2]}}$.

The angle $\\theta$ between any two vectors $\\boldsymbol{a}$ and $\\boldsymbol{b}$ can be calculated using

\\[\\cos(\\theta)=\\frac{\\boldsymbol{a\\cdot b}}{\\lvert\\boldsymbol{a}\\rvert\\lvert\\boldsymbol{b}\\rvert},\\]

where $\\lvert\\boldsymbol{x}\\rvert=\\sqrt{x_1^2+x_2^2+x_3^2}$ is the length of the vector $\\boldsymbol{x}$.

The angle $\\theta$ between the tangent vectors at the point $\\boldsymbol{p}$ is the angle between the vectors $\\boldsymbol{u}$ and $\\boldsymbol{v}$, so

\\[\\cos(\\theta)=\\frac{(\\var{u[0]}\\times\\var{v[0]})+(\\var{u[1]}\\times\\var{v[1]})+(\\var{u[2]}\\times\\var{v[2]})}{\\sqrt{(\\var{u[0]})^2+(\\var{u[1]})^2+(\\var{u[2]})^2}\\sqrt{(\\var{v[0]})^2+(\\var{v[1]})^2+(\\var{v[2]})^2}}=\\frac{\\var{dotuv}}{\\var{precround(lenu,4)}\\times\\var{precround(lenv,4)}}=\\var{precround(dotuv/(lenu*lenv),4)}\\;\\text{to 4d.p.}\\]

Then $\\theta=\\arccos(\\var{precround(dotuv/(lenu*lenv),4)})=\\var{theta}^\\circ$ to 2d.p.

In an identical way, the angle $\\phi$ between the tangent vectors at the point $\\boldsymbol{q}$ is the angle between the vectors $\\boldsymbol{u}$ and $\\boldsymbol{w}$, so $\\phi=\\var{phi}^\\circ$ to 2d.p.

", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"lenu": {"name": "lenu", "group": "Ungrouped variables", "definition": "abs(u)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "vector(a1*t,b1*t,c1*t)", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "vector(d1,0,0)", "description": "", "templateType": "anything", "can_override": false}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "vector(d1,2*e1*tau,3*f1*tau^2)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "(e1*a1^2)*t/d1^2", "description": "", "templateType": "anything", "can_override": false}, "phi": {"name": "phi", "group": "Ungrouped variables", "definition": "precround(degrees(arccos(dotuw/(lenu*lenw))),2)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "dotuv": {"name": "dotuv", "group": "Ungrouped variables", "definition": "dot(u,v)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "dotuw": {"name": "dotuw", "group": "Ungrouped variables", "definition": "dot(u,w)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "(f1*d1*b1^2)/(a1*e1^2)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "lenw": {"name": "lenw", "group": "Ungrouped variables", "definition": "abs(w)", "description": "", "templateType": "anything", "can_override": false}, "theta": {"name": "theta", "group": "Ungrouped variables", "definition": "precround(degrees(arccos(dotuv/(lenu*lenv))),2)", "description": "", "templateType": "anything", "can_override": false}, "lenv": {"name": "lenv", "group": "Ungrouped variables", "definition": "abs(v)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "d1*random(-2..2 except 0)", "description": "", "templateType": "anything", "can_override": false}, "tau": {"name": "tau", "group": "Ungrouped variables", "definition": "a1*t/d1", "description": "", "templateType": "anything", "can_override": false}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "vector(a1,b1,c1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "phi", "lenu", "dotuw", "tau", "e1", "dotuv", "a1", "u", "t", "w", "v", "lenw", "lenv", "d1", "q", "theta", "c1", "b1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Enter the least value of $t$, and the corresponding value of $\\tau$, defining the first intersection point. Hence enter the values of the intersection point $\\boldsymbol{p}$ for these values of $t$ and $\\tau$.

$t=$ [[0]]; $\\tau=$ [[1]].

$\\boldsymbol{p}=($[[2]]$,$[[3]]$,$[[4]]$)$.

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Enter the greatest value of $t$, and the corresponding value of $\\tau$, defining the second intersection point. Hence enter the values of the intersection point $\\boldsymbol{q}$ for these values of $t$ and $\\tau$.

$t=$ [[0]]; $\\tau=$ [[1]].

$\\boldsymbol{q}=($[[2]]$,$[[3]]$,$[[4]]$)$.

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "t", "maxValue": "t", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tau", "maxValue": "tau", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q[0]", "maxValue": "q[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q[1]", "maxValue": "q[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q[2]", "maxValue": "q[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the tangent vector $\\boldsymbol{u}$ of the curve $\\mathcal{C}_1$.

$\\boldsymbol{u}=($[[0]]$,$[[1]]$,$[[2]]$)$.

Find the tangent vector $\\boldsymbol{v}$ of the curve $\\mathcal{C}_2$ at the point $\\boldsymbol{p}$.

$\\boldsymbol{v}=($[[0]]$,$[[1]]$,$[[2]]$)$.

Find the tangent vector $\\boldsymbol{w}$ of the curve $\\mathcal{C}_2$ at the point $\\boldsymbol{q}$.

$\\boldsymbol{w}=($[[0]]$,$[[1]]$,$[[2]]$)$.

Calculate the angle $\\theta$ (in degrees) between the tangent vectors of each curve, at the point $\\boldsymbol{p}$.

$\\theta=$ [[0]]$^\\circ$. (Enter your answer to 2d.p.)

Calculate the angle $\\phi$ (in degrees) between the tangent vectors of each curve, at the point $\\boldsymbol{q}$.

$\\phi=$ [[0]]$^\\circ$. (Enter your answer to 2d.p.)