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Find the angle between $\\boldsymbol{v}$ and $\\boldsymbol{w}$, in radians.

\n

Note the angle must be in the range $0$ to $\\pi$.

\n

Give your answer to {precision} decimal places.

\n

Angle in radians = [[0]]

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You are given the vectors $\\boldsymbol{v} = \\var{v}$, $\\boldsymbol{w} = \\var{w}$ in $\\mathbb{R}^3$.

", "tags": ["angle between vectors", "angle beween two vectors", "checked2015", "degrees and radians", "dot product", "finding the angle between vectors", "inner product", "MAS1602", "mas1602", "radians", "scalar product", "vectors"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

15/7/2015

\n

Added tags

\n

\n

16/07/2012:

\n

Added tags.

\n

Question appears to be working correctly.

\n

Moved the \\rightarrow to the correct place in the solution.

\n

 

\n

 

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given vectors  $\\boldsymbol{v,\\;w}$, find the angle between them.

"}, "advice": "

Use the formula, $\\boldsymbol{v \\cdot w} = \\lVert \\boldsymbol{v} \\rVert \\lVert \\boldsymbol{w} \\rVert \\cos(\\theta)$m where $\\theta$ is the angle between the vectors.

\n

Here

\n

\\begin{align}
\\lVert \\boldsymbol{v} \\rVert &= \\simplify[]{sqrt({s1}^2 + {s2}^2)} \\\\
&= \\sqrt{2}, \\\\[1em]
\\lVert \\boldsymbol{w} \\rVert &= \\simplify[]{sqrt({s3}^2 + {s4}^2)} \\\\
&= \\sqrt{2}, \\\\[1em]
\\boldsymbol{v \\cdot w} &= \\var{v} \\boldsymbol{\\cdot} \\var{w} \\\\
&= \\var{dot(v,w)}
\\end{align}

\n

So

\n

\\begin{align}
\\cos(\\theta) &= \\frac{\\var{dot(v,w)}}{\\sqrt{2}\\sqrt{2}} = \\simplify[std]{{dot(v,w)}/2} \\\\
\\implies \\theta &= \\arccos\\left(\\simplify[std]{{dot(v,w)}/{2}}\\right) \\\\
&= \\var{precround(angle,precision)} \\text{ radians}
\\end{align}

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We can write a vector equation of the plane in the form:

\n

$\\boldsymbol{r}=\\boldsymbol{r_1}+\\lambda (\\boldsymbol{r_2}-\\boldsymbol{r_1}) + \\mu (\\boldsymbol{r_3}-\\boldsymbol{r_1})$

\n

Note that three points determine a plane and

\n
    \n
  1. $\\lambda=0,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_1}$
  2. \n
  3. $\\lambda=1,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_2}$
  4. \n
  5. $\\lambda=0,\\;\\;\\mu=1$ gives $\\boldsymbol{r}=\\boldsymbol{r_3}$
  6. \n
\n

Note that if we let

\n

\\[\\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1})\\]

\n

then $\\boldsymbol{n}\\cdot (\\boldsymbol{r_2}-\\boldsymbol{r_1})=0$ and $\\boldsymbol{n}\\cdot (\\boldsymbol{r_3}-\\boldsymbol{r_1})=0$.

\n

If $\\boldsymbol{r} = (x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that

\n

\\[ \\boldsymbol{r}\\cdot \\boldsymbol{n}=(x,\\;y,\\;z)\\cdot  \\boldsymbol{n}=\\boldsymbol{r_1}\\cdot \\boldsymbol{n} \\]

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Find the Cartesian equation of this plane, in the form $ax+by+cz = d$, with $a$, $b$ and $c$ integers, not decimals.

\n

Equation of the plane: [[0]] $ = $ [[1]]

\n

You can get help by clicking on Show steps.

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Input numbers as integers and not decimals

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Normal to the plane

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A plane goes through the points:

\n

\\begin{align}
\\boldsymbol{r_1} &= \\var{r_1}, & \\boldsymbol{r_2} &= \\var{r_2}, & \\boldsymbol{r_3} &= \\var{r_3}
\\end{align}

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A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.

\n

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.

"}, "advice": "

We can write a vector equation of the plane in the form:

\n

$\\boldsymbol{r}=\\boldsymbol{r_1}+\\lambda (\\boldsymbol{r_2}-\\boldsymbol{r_1}) + \\mu (\\boldsymbol{r_3}-\\boldsymbol{r_1})$

\n

Note that three points determine a plane and

\n
    \n
  1. $\\lambda=0,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_1}$
  2. \n
  3. $\\lambda=1,\\;\\;\\mu=0$ gives $\\boldsymbol{r}=\\boldsymbol{r_2}$
  4. \n
  5. $\\lambda=0,\\;\\;\\mu=1$ gives $\\boldsymbol{r}=\\boldsymbol{r_3}$
  6. \n
\n

Note that if we let

\n

\\[\\boldsymbol{n}=(\\boldsymbol{r_2}-\\boldsymbol{r_1})\\times (\\boldsymbol{r_3}-\\boldsymbol{r_1})\\]

\n

then $\\boldsymbol{n}\\cdot (\\boldsymbol{r_2}-\\boldsymbol{r_1})=0$ and $\\boldsymbol{n}\\cdot (\\boldsymbol{r_3}-\\boldsymbol{r_1})=0$.

\n

Hence $\\boldsymbol{r}\\cdot \\boldsymbol{n}=\\boldsymbol{r_1}\\cdot \\boldsymbol{n}$.

\n

If $\\boldsymbol{r} = (x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that

\n

\\[ \\boldsymbol{r}\\cdot \\boldsymbol{n}=(x,\\;y,\\;z)\\cdot  \\boldsymbol{n}=\\boldsymbol{r_1}\\cdot \\boldsymbol{n} \\]

\n

If $\\boldsymbol{r}=(x,\\;y,\\;z)$ are the Cartesian coordinates of a point on the line, it follows that the equation of the plane is given by $\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} \\cdot \\boldsymbol{n} = \\boldsymbol{r_1} \\cdot \\boldsymbol{n}$.

\n

We have:

\n

\\[ \\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} \\]

\n

Hence, $\\boldsymbol{r_1} \\cdot \\boldsymbol{n} = \\var{con}$.

\n

So the Cartesian equation of the plane is

\n

\\[ \\simplify[all,!noLeadingMinus]{{coeffx}x + {coeffy}y + {coeffz}z = {con}} \\]

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The vector equation of a plane is

\n

\\[ \\boldsymbol{x}=\\boldsymbol{x_0}+\\lambda \\boldsymbol{v} + \\mu \\boldsymbol{w} \\]

\n

If you let $\\boldsymbol{n}=\\boldsymbol{v} \\times \\boldsymbol{w}$, then $\\boldsymbol{x} \\cdot \\boldsymbol{n} = \\boldsymbol{x_0} \\cdot \\boldsymbol{n}$ as $\\boldsymbol{v}\\cdot \\boldsymbol{n}=0$ and $\\boldsymbol{w}\\cdot \\boldsymbol{n}=0$.

\n

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:

\n

\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} \\cdot \\boldsymbol{n} =\\boldsymbol{x_0} \\cdot \\boldsymbol{n}\\]

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Enter $ax + by + cz$ with $a$, $b$ and $c$ integers, not decimals.

\n

Equation of the plane: [[0]] $ = $ [[1]]

\n

You can get help by clicking on Show steps.

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Input numbers as integers and not decimals

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You are given the vector equation of a plane in $\\mathbb{R}^3$:

\n

\\[ \\boldsymbol{x} = \\var{x_0} + \\lambda \\var{v} + \\mu \\var{w}, \\quad -\\infty\\lt\\lambda,\\;\\mu \\lt \\infty \\]

\n

In this question you want to find an equation of this plane in the Cartesian from

\n

\\[ ax + by + cz = d \\]

", "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"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find the Cartesian form $ax+by+cz=d$ of the equation of the plane $\\boldsymbol{r=r_0+\\lambda a+\\mu b}$.

\n

The solution is not unique. The constant on right hand side could be given to ensure that the left hand side is unique.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The vector equation of the plane is

\n

\\[ \\boldsymbol{x}=\\boldsymbol{x_0}+\\lambda \\boldsymbol{v} + \\mu \\boldsymbol{w} \\]

\n

where

\n

\\begin{align}
\\boldsymbol{x_0} &= \\var{x_0}, & \\boldsymbol{v} &= \\var{v}, & \\boldsymbol{w} &= \\var{w}
\\end{align}

\n

We have

\n

\\[ \\boldsymbol{n} = \\boldsymbol{v} \\times \\boldsymbol{w} = \\var{v} \\times \\var{w} = \\var{cross(v,w)} \\]

\n

If you let $\\boldsymbol{n}=\\boldsymbol{v} \\times \\boldsymbol{w}$, then $\\boldsymbol{x} \\cdot \\boldsymbol{n} = \\boldsymbol{x_0} \\cdot \\boldsymbol{n}$ as $\\boldsymbol{v}\\cdot \\boldsymbol{n}=0$ and $\\boldsymbol{w}\\cdot \\boldsymbol{n}=0$.

\n

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:

\n

\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} \\cdot \\boldsymbol{n} =\\boldsymbol{x_0} \\cdot \\boldsymbol{n}\\]

\n

That is,

\n

\\[ \\simplify[all,!noLeadingMinus]{ {coeffx}*x+{coeffy}*y + {coeffz}*z } = \\var{con} \\]

"}, {"name": "Determine if vectors are perpendicular", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(c,d,f)", "description": "", "name": "w"}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(a,b,lambda)", "description": "", "name": "v"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,4,5,10)", "description": "", "name": "f"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "lambda": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(-a*c-b*d)/f", "description": "", "name": "lambda"}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s5"}, "mu1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "lcm(random(-5..5 except 0),f)", "description": "", "name": "mu1"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "mu2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "lcm(random(-5..5 except 0),f)", "description": "", "name": "mu2"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "d"}, "u": {"group": "Ungrouped variables", "templateType": "anything", "definition": "mu1*v+mu2*w", "description": "", "name": "u"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "lambda", "mu1", "mu2", "v", "w", "u"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

When are vectors $\\boldsymbol{v,\\;w}$ orthogonal?

\n

Part b) is not answered in Advice, the given solution is for a different question.

"}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find $\\lambda \\in \\mathbb{R}$ such that $\\boldsymbol{v}$ and $\\boldsymbol{w}$ are orthogonal.

\n

$\\lambda = $ [[0]]

", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "lambda", "maxValue": "lambda", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "variableReplacements": [], "marks": 1.5, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

Find $\\lambda \\in \\mathbb{R}$ such that the vector $\\boldsymbol{u} = \\simplify[fractionnumbers]{{u}}$ is contained in the plane through the origin parallel to $\\boldsymbol{v}$ and $\\boldsymbol{w}$.

\n

$\\lambda =$ [[0]]

", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "lambda", "maxValue": "lambda", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "variableReplacements": [], "marks": 0.5, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "statement": "

You are given the vectors $\\boldsymbol{v} = \\begin{pmatrix}\\var{a} \\\\ \\var{b} \\\\ \\lambda \\end{pmatrix}$ and $\\boldsymbol{w} = \\begin{pmatrix} \\var{c} \\\\ \\var{d} \\\\ \\var{f} \\end{pmatrix}$.

\n

Enter your answers to the following questions as fractions or integers, not decimals.

", "tags": ["checked2015", "dot product", "finding perpendicular vectors", "inner product", "perpendicular vectors", "product", "scalar product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "variablesTest": {"condition": "u<>vector(0,0,0)", "maxRuns": 100}, "advice": "

a)

\n

$\\boldsymbol{v}$ and $\\boldsymbol{w}$ are perpendicular to one another when $\\boldsymbol{v} \\cdot \\boldsymbol{w} = 0$.

\n

Now

\n

\\begin{align}
\\boldsymbol{v} \\cdot \\boldsymbol{w} &= \\simplify[]{{a}*{c}+{b}*{d}+lambda*{f}} \\\\
&= \\simplify[std]{{f}*lambda+{a*c+b*d}}
\\end{align}

\n

Hence

\n

\\[\\boldsymbol{v} \\cdot \\boldsymbol{w} = 0 \\implies \\simplify[std]{{f}*lambda+{a*c+b*d}}=0 \\implies \\lambda = \\simplify[std]{{-a*c-b*d}/{f}}\\]

\n

b)

\n

$\\boldsymbol{v}$ is in the $xy$ plane when $\\lambda=0$.

"}, {"name": "Dot and cross product combinations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"displayType": "radiogroup", "layout": {"type": "all", "expression": ""}, "choices": ["$\\boldsymbol{(v\\cdot w)\\cdot u}$", "$\\boldsymbol{(v\\cdot w)u}$", "$\\boldsymbol{(v\\cdot w)\\times u}$", "$\\boldsymbol{(v\\times w)\\times u}$", "

$\\boldsymbol{(v\\times w)\\cdot u}$

"], "variableReplacementStrategy": "originalfirst", "matrix": [[0, 0, 0.4], [0, 0.4, 0], [0, 0, 0.4], [0, 0.4, 0], [0.4, 0, 0]], "shuffleChoices": true, "type": "m_n_x", "maxAnswers": 0, "marks": 0, "warningType": "none", "scripts": {}, "minMarks": 0, "minAnswers": 0, "maxMarks": 0, "shuffleAnswers": true, "showCorrectAnswer": true, "variableReplacements": [], "answers": ["

Scalar

", "

Vector

", "

Undefined

"]}], "variables": {}, "ungrouped_variables": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Given the vectors $\\boldsymbol{v}$, $\\boldsymbol{w}$, $\\boldsymbol{u}$ in $\\mathbb{R}^3$, state whether the following quantities are scalars (real numbers), vectors (elements of $\\mathbb{R}^3$) or undefined.

\n

In this question, the symbol $\\cdot$ denotes the inner product and $\\times$ always denotes the cross product.

", "tags": ["checked2015", "cross product", "dot product", "inner product", "MAS1602", "mas1602", "scalar product", "scalars", "unused", "vector", "Vector", "vector product", "vectors"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

15/07/2012:

\n \t\t

Added tags.

\n \t\t

16/07/2012:

\n \t\t

Added tags.

\n \t\t

 

\n \t\t

 

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Determine if various combinations of vectors are defined or not.

"}, "advice": "

1. $\\boldsymbol{(v\\cdot w)\\cdot u}$ is undefined as $\\boldsymbol{v\\cdot w}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{u}$.

\n

2. $\\boldsymbol{(v\\cdot w) u}$ is a vector and is a scalar multiple of $\\boldsymbol{u}$ as $\\boldsymbol{v \\cdot w}$ is a scalar.

\n

3. $\\boldsymbol{(v \\cdot w)\\times u}$ is undefined as $\\boldsymbol{v\\cdot w}$ is a scalar and the cross product is only defined between vectors.

\n

4. $\\boldsymbol{(v\\times w)\\times u}$ is a vector as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the cross product between vectors produces a vector.

\n

5. $\\boldsymbol{(v\\times w)\\cdot u}$ is a scalar as $\\boldsymbol{v \\times w}$ and $\\boldsymbol{u}$ are vectors and the inner or dot product is between vectors and produces a scalar.

"}, {"name": "Find the angle between planes", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

$\\alpha = $ [[0]] radians

\n

(Enter your answer in radians, to 3 decimal places)

", "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"], "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Calculate the angle $\\alpha,\\;0\\leq\\alpha \\leq \\frac{\\pi}{2}$, between the plane $\\Pi_1$, passing through the points

\n

\\begin{align}
\\boldsymbol{r_1} &= \\var{r1}, & \\boldsymbol{r_2} &= \\var{r2}, & \\boldsymbol{r_3} &= \\var{r3}
\\end{align}

\n

and the plane, $\\Pi_2$, whose equation is

\n

\\[\\simplify[std]{{a1}x+{b1}y+{c1}z={d1}}\\]

", "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", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find angle between plane $\\Pi_1$, given by three points, and the plane $\\Pi_2$ given in Cartesian form.

\n

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.

"}, "advice": "

The angle between two planes is given by the angle between their normals.

\n

The plane $\\Pi_1$ can be written in the form

\n

\\[\\boldsymbol{r} = \\boldsymbol{r}_1+\\lambda( \\boldsymbol{r}_2 - \\boldsymbol{r}_1)+\\mu( \\boldsymbol{r}_3- \\boldsymbol{r}_1)\\]

\n

and the normal $\\boldsymbol{n}_1$ to this plane is given by:

\n

\\[ \\boldsymbol{n}_1 = (\\boldsymbol{r}_2 - \\boldsymbol{r}_1)\\times (\\boldsymbol{r}_3- \\boldsymbol{r}_1)\\]

\n

For $ \\boldsymbol{r}_1$, $\\boldsymbol{r}_2$, $\\boldsymbol{r}_3$ as given.

\n

\\[ \\boldsymbol{n}_1 = \\var{n1} \\]

\n

The normal to the plane $\\Pi_2$ is given by

\n

\\[ \\boldsymbol{n}_2 = \\var{vector(a1,b1,c1)} \\]

\n

The angle between the two normals (and hence the two planes) can be found using:

\n

\\[ \\cos(\\alpha) = \\frac{\\boldsymbol{n}_1 \\cdot \\boldsymbol{n}_2}{\\lVert\\boldsymbol{n}_1\\rVert \\lVert\\boldsymbol{n}_2\\rVert} \\]

\n

On calculating this, we obtain

\n

\\begin{align}
\\boldsymbol{n}_1 \\boldsymbol{\\cdot} \\boldsymbol{n}_2 &= \\var{a1*n1[0]+b1*n1[1]+c1*n1[2]} \\\\
\\lVert\\boldsymbol{n}_1\\rVert &= \\simplify[std]{sqrt({n1[0]^2+n1[1]^2+n1[2]^2})} \\\\
\\lVert\\boldsymbol{n}_2\\rVert &= \\simplify[std]{sqrt({a1^2+b1^2+c1^2})}\\\\
\\cos(\\alpha) &= \\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))}}
\\end{align}

\n

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 calculated value from $\\pi$ to obtain an acute angle. So the angle between the two planes is 

\n

\\[ \\alpha' = \\pi - \\var{precround(alpha,3)} = \\var{precround(ans,3)} \\text{ radians} \\]

\n

\\[ \\alpha = \\var{precround(ans,3)} \\text{ radians} \\]

\n

to 3 decimal places.

\n

"}, {"name": "Inner product of two vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "{inner}", "maxValue": "{inner}", "variableReplacements": [], "marks": 2}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

Find $\\boldsymbol{v \\cdot w} = $ [[0]]

", "variableReplacements": [], "marks": 0}], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s5", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "name": "c", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "f", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s4", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}, "inner": {"group": "Ungrouped variables", "templateType": "anything", "definition": "{a*c+b*d+f*g}", "name": "inner", "description": ""}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s3", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "a", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "name": "d", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "g", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s3", "s2", "s1", "s5", "s4", "inner"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "variable_groups": [], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

You are given the vectors $\\boldsymbol{v}= \\var{vector(a,b,g)}$ and $\\boldsymbol{w} = \\var{vector(c,d,f)}$ in $\\mathbb{R}^3$.

", "tags": ["checked2015", "dot product", "dot product of two vectors", "inner product", "mas1602", "MAS1602", "scalar product", "three dimensional vectors", "unused", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

15/07/2012:

\n \t\t

Added tags.

\n \t\t

16/07/2012:

\n \t\t

Added tags.

\n \t\t

Question appears to be working correctly.

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given vectors $\\boldsymbol{v}$ and $\\boldsymbol{w}$, find their inner product.

"}, "advice": "

\\begin{align}
\\boldsymbol{v \\cdot w} &= \\var{vector(a,b,g)} \\boldsymbol{\\cdot} \\var{vector(c,d,f)} \\\\
&= \\simplify[]{{a}*{c}+{b}*{d}+{g}*{f}} \\\\
&= \\var{inner}
\\end{align}

"}, {"name": "Minimum distance between a point and a line in 3D", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "variable_groups": [{"variables": ["directions", "p"], "name": "Vector p"}, {"variables": ["s1", "a", "b", "g", "x_0"], "name": "Vector x_0"}, {"variables": ["c", "d", "f", "v"], "name": "Vector v"}, {"variables": ["top"], "name": "Answer"}], "variables": {"s1": {"group": "Vector x_0", "templateType": "anything", "definition": "random(-1,1)", "description": "", "name": "s1"}, "v": {"group": "Vector v", "templateType": "anything", "definition": "vector(random(-9..9 except -1..1),random(-9..9 except -1..1),random(2..9))", "description": "", "name": "v"}, "d": {"group": "Vector v", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "d"}, "top": {"group": "Answer", "templateType": "anything", "definition": "cross(p-x_0,v)", "description": "

Top of the formula for the minimum distance: $(p-x_0) \\times v$.

", "name": "top"}, "f": {"group": "Vector v", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "f"}, "directions": {"group": "Vector p", "templateType": "anything", "definition": "map(id(3)[x],x,shuffle(0..2))", "description": "

Shuffled list of axis vectors

", "name": "directions"}, "b": {"group": "Vector x_0", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "b"}, "a": {"group": "Vector x_0", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "g": {"group": "Vector x_0", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}, "p": {"group": "Vector p", "templateType": "anything", "definition": "random(-1,1)*directions[0] + random(-5..5 except -1..1)*directions[1]", "description": "", "name": "p"}, "x_0": {"group": "Vector x_0", "templateType": "anything", "definition": "vector(random(2..9),0,random(2..9))*random(-1,1) + vector(0,random(2..9),0)*random(-1,1)", "description": "", "name": "x_0"}, "c": {"group": "Vector v", "templateType": "anything", "definition": "random(-1,1)*random(2..9)", "description": "", "name": "c"}}, "ungrouped_variables": [], "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "showCorrectAnswer": true, "unitTests": [], "prompt": "

The minimum distance between the line and the point is given by

\n

\\[ \\frac{\\left\\lVert(\\boldsymbol{p} - \\boldsymbol{x_0})\\times \\boldsymbol{v} \\right\\rVert}{\\lVert \\boldsymbol{v} \\rVert}\\]

", "variableReplacements": [], "marks": 0}], "prompt": "

Distance = [[0]]

\n

Enter your answer exactly, using the function sqrt(x) if necessary. Do not use decimals.

\n

You can get help by clicking on Show steps.

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "sqrt({abs(top)^2}/{abs(v)^2})", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "showPreview": true, "notallowed": {"message": "

Enter all numbers as integers, do not use decimals

", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "unitTests": [], "checkVariableNames": false, "vsetRange": [0, 1], "vsetRangePoints": 5, "failureRate": 1, "scripts": {}, "answerSimplification": "std", "type": "jme", "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

In $\\mathbb{R}^3$ find the distance between the point $\\boldsymbol{p} = \\var{p}$ and the line through the point $\\boldsymbol{x_0} = \\var{x_0}$ that is parallel to the vector $\\boldsymbol{v} = \\var{v}$.

", "tags": ["checked2015", "cross product of vectors", "distance between a point and a line", "distance between two points", "equation of a line through a point and in the direction of a vector", "minimum distance", "minimum distance between a point and a line in three space", "modulus of a vector", "three dimensional vector geometry", "vector", "Vector", "vector geometry", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find minimum distance between a point and a line in 3-space. The line goes through a given point in the direction of a given vector.

\n

The correct solution is given, however the accuracy of 0.001  is not enough as in some cases answers near to the correct solution are also marked as correct.

"}, "advice": "

The line through $\\boldsymbol{x_0} = \\var{x_0}$ in the direction of $\\boldsymbol{v}=\\var{v}$ has equation:

\n

\\[ \\boldsymbol{r} = \\boldsymbol{x_0} + \\lambda \\boldsymbol{v} = \\var{x_0} + \\lambda \\var{v} \\]

\n

The minimum distance between this line and the point $\\boldsymbol{p} = \\var{p}$ is given by

\n

\\[ \\frac{\\left\\lVert(\\boldsymbol{p}-\\boldsymbol{x_0}) \\times \\boldsymbol{v} \\right\\rVert}{\\lVert \\boldsymbol{v} \\rVert} \\]

\n

Now,

\n

\\begin{align}
\\boldsymbol{p} - \\boldsymbol{x_0} &= \\var{p-x_0} \\Rightarrow \\\\[1em]
(\\boldsymbol{p}-\\boldsymbol{x_0}) \\times \\boldsymbol{v} &= \\var{p-x_0} \\times \\var{v} \\\\[1em]
&= \\var{top}
\\end{align}

\n

Since

\n

\\begin{align}
\\left\\lVert \\begin{matrix} \\var{top[0]} \\\\ \\var{top[1]} \\\\ \\var{top[2]} \\end{matrix} \\right \\rVert &= \\simplify[]{sqrt({top[0]}^2 + {top[1]}^2 + {top[2]}^2)} \\\\ &= \\sqrt{\\var{abs(top)^2}}
\\end{align}

\n

and $\\lVert \\boldsymbol{v} \\rVert = \\simplify[]{sqrt({v[0]}^2 + {v[1]}^2 + {v[2]}^2)} = \\sqrt{\\var{abs(v)^2}}$, the distance is then:

\n

\\[\\simplify{sqrt({abs(top)^2}/{abs(v)^2})}\\]

"}, {"name": "Scalar triple product of coplanar vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [{"variables": ["w1", "w2", "w3"], "name": "w"}, {"variables": ["s1", "s2", "s3", "s4", "s5", "s6"], "name": "s"}, {"variables": ["x1", "x2", "x3"], "name": "x"}, {"variables": ["y1", "y2", "y3"], "name": "y"}, {"variables": ["z1", "z2"], "name": "z"}], "variables": {"s1": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "x2": {"group": "x", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "x2"}, "y3": {"group": "y", "templateType": "anything", "definition": "if(x1=x3,if(y1=ty3,-ty3,ty3),ty3)", "description": "", "name": "y3"}, "w1": {"group": "w", "templateType": "anything", "definition": "x2*y3-y2*x3", "description": "", "name": "w1"}, "s4": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "y2": {"group": "y", "templateType": "anything", "definition": "if(x1*z2=x2*y1,z2+1,z2)", "description": "", "name": "y2"}, "s2": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "z2": {"group": "z", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "z2"}, "x3": {"group": "x", "templateType": "anything", "definition": "random(-9..9)", "description": "", "name": "x3"}, "lambda": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(x3*(y1*z2-y2*z1)+y3*(x2*z1-x1*z2))/(x2*y1-x1*y2)", "description": "", "name": "lambda"}, "w2": {"group": "w", "templateType": "anything", "definition": "x3*y1-y3*x1", "description": "", "name": "w2"}, "x1": {"group": "x", "templateType": "anything", "definition": "s1*random(1..9)", "description": "", "name": "x1"}, "s5": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s5"}, "s3": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "s6": {"group": "s", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s6"}, "y1": {"group": "y", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "y1"}, "z1": {"group": "z", "templateType": "anything", "definition": "s5*random(1..9)", "description": "", "name": "z1"}, "w3": {"group": "w", "templateType": "anything", "definition": "x1*y2-x2*y1", "description": "", "name": "w3"}, "ty3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s6*random(1..9)", "description": "", "name": "ty3"}}, "ungrouped_variables": ["ty3", "lambda"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "allowFractions": true, "type": "numberentry", "showCorrectAnswer": true, "minValue": "lambda", "correctAnswerFraction": true, "variableReplacements": [], "marks": 2, "maxValue": "lambda"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "

$\\lambda=$ [[0]].

\n

Enter your answer as a fraction or integer and not a decimal.

", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

You are given three points in $\\mathbb{R}^3$,

\n

\\begin{align}
\\boldsymbol{a} &= \\var{vector(x1,x2,x3)}, &
\\boldsymbol{b} &= \\var{vector(y1,y2,y3)}, &
\\boldsymbol{c} &= \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix}
\\end{align}

\n

where $\\lambda$ is a parameter to be determined.

\n

Find the value of $\\lambda$ such that $\\boldsymbol{a}$, $\\boldsymbol{b}$ and $\\boldsymbol{c}$ all lie on the same plane through the origin.

", "tags": ["checked2015", "colinear", "colinearvectors", "cross product", "determining if three vectors in three space are colinear", "dot product", "inner product", "scalar product", "scalar triple product", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

14/7/2015

\n

Added module tag

\n

\n

03/12/2013

\n

Clarified wording vector -> position vector. (AJY)

\n

Typo colinear -> collinear. (AJY)

\n

15/07/2012:

\n

Added tags.

\n

Corrected error. It was possible for the third coordinate of $A \\times B$  to be 0. Hence could not be colinear in some circumstances. Calculations checked as a result.

\n

16/07/2012:

\n

Added tags.

\n

Moved \\rightarrow so that it is located at the beginning of the line.

\n

Question appears to be working correctly.

\n

27/08/2012:

\n

Changed coplanar to colinear - for obvious reasons!!

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Three 3 dim vectors, one with a parameter $\\lambda$ in the third coordinate. Find value of $\\lambda$ ensuring vectors coplanar. Scalar triple product.

"}, "advice": "

Note that $\\boldsymbol{a}\\times \\boldsymbol{b}$ is a vector which is perpendicular to both $\\boldsymbol{a}$ and $\\boldsymbol{b}$ and hence to the plane through the origin containing $\\boldsymbol{a}$ and $\\boldsymbol{b}$. 

\n

So if $\\boldsymbol{c}$ is perpendicular to $\\boldsymbol{a} \\times \\boldsymbol{b}$, i.e. $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} = 0$, it must lie on the same plane.

\n

Now

\n

\\begin{align} 
\\boldsymbol{a} \\times \\boldsymbol{b} &= \\var{vector(x1,x2,x3)} \\times \\var{vector(y1,y2,y3)} \\\\[1em]
&= \\simplify[]{vector({x2}*{y3}-{x3}*{y2}, {x3}*{y1}-{x1}*{y3}, {x1}*{y2}-{x2}*{y1})} \\\\[1em]
&= \\var{vector(w1,w2,w3)}
\\end{align}

\n

Hence

\n

\\begin{align}
(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &= \\var{vector(w1,w2,w3)} \\boldsymbol{\\cdot} \\begin{pmatrix} \\var{z1} \\\\ \\var{z2} \\\\ \\lambda \\end{pmatrix} \\\\[1em]
&= \\simplify[]{{w1}*{z1}+{w2}*{z2}+{w3}*lambda} \\\\[1em]
&= \\simplify{{w1*z1+w2*z2}+{w3}*lambda}
\\end{align}

\n


We now require a value of $\\lambda$ so that $(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c}=0$.

\n

\\begin{align}
&&(\\boldsymbol{a}\\times \\boldsymbol{b})\\boldsymbol{\\cdot} \\boldsymbol{c} &=  0 \\\\
\\implies &&\\simplify{{w1*z1+w2*z2}+{w3}*lambda} &= 0 \\\\
\\implies &&\\lambda &= \\simplify[std]{{-w1*z1-w2*z2}/{w3}}
\\end{align}

"}, {"name": "Vector cross product", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s1"}, "result": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(vector(a,b,g),vector(c,d,f))", "description": "", "name": "result"}, "s5": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s5"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "description": "", "name": "c"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "description": "", "name": "f"}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s4"}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "a"}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s3"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "description": "", "name": "d"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "description": "", "name": "g"}}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "result", "s1", "s2", "s3", "s4", "s5"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "

Find

\n

$\\boldsymbol{v} \\times \\boldsymbol{w} = $ [[0]]

", "scripts": {}, "gaps": [{"allowFractions": false, "correctAnswer": "result", "showCorrectAnswer": true, "allowResize": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst", "numRows": "3", "scripts": {}, "type": "matrix", "numColumns": 1, "tolerance": 0, "markPerCell": false, "variableReplacements": [], "marks": "3"}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

You are given the vectors $\\boldsymbol{v} = \\var{vector(a,b,g)}$, $\\boldsymbol{w} = \\var{vector(c,d,f)}$.

", "tags": ["3 dimensional vector", "checked2015", "cross product", "three dimensional vectors", "unused", "Vector", "vector", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

14/7/2015

\n

Added unused tag

\n

\n

16/07/2012:

\n

Added tags.

\n

Question appears to be working correctly.

\n

 

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Given vectors $\\boldsymbol{A,\\;B}$, find $\\boldsymbol{A\\times B}$

"}, "advice": "

\\begin{align}
\\boldsymbol{v} \\times \\boldsymbol{w} &= \\begin{pmatrix} \\simplify[basic]{{b}*{f}-{g}*{d}} \\\\ \\simplify[basic]{{g}*{c}-{a}*{f}} \\\\ \\simplify[basic]{{a}*{d}-{b}*{c}}  \\end{pmatrix} \\\\[1em]
&= \\var{result}
\\end{align}

"}, {"name": "Vector equation of a line, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "parts": [{"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

Find the vector equation of Line 1 which goes through the point $\\boldsymbol{x_0}$ in the direction of the vector $\\boldsymbol{v}$.

\n

Input the vector equation in the form:

\n

\\[\\boldsymbol{r} = \\begin{pmatrix} a_1 \\\\ a_2 \\\\ a_3 \\end{pmatrix} + \\lambda \\begin{pmatrix} b_1 \\\\ b_2 \\\\ b_3 \\end{pmatrix} \\]

\n

such that $\\boldsymbol{r} = \\boldsymbol{x_0}$ when $\\lambda=0$ and $\\boldsymbol{r}=\\boldsymbol{x_0}+\\boldsymbol{v}$ when $\\lambda=1$ by filling in the appropriate fields below:

\n

$ \\boldsymbol{r} = $ [[0]] $ + \\lambda $ [[1]]

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(a,b,g)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(c,d,f)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

Once again find the vector equation of Line 2 which goes through the point $\\boldsymbol{y_0}$ in the direction of the vector $\\boldsymbol{w}$ in the form

\n

\\[ \\boldsymbol{r} = \\begin{pmatrix} c_1 \\\\ c_2 \\\\ c_3 \\end{pmatrix} + \\mu \\begin{pmatrix} d_1 \\\\ d_2 \\\\ d_3 \\end{pmatrix} \\]

\n

such that $\\boldsymbol{r}=\\boldsymbol{C}$ when $\\mu=0$ and $\\boldsymbol{r=C+D}$ when $\\mu=1$ by filling in the appropriate fields below:

\n

$ \\boldsymbol{r} = $ [[0]] $ + \\mu $ [[1]]

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(a1,b1,g1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}, {"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "vector(c1,d1,f1)", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "0.75", "numRows": "3"}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "

You are told that Line 1 and Line 2 intersect in a point $\\boldsymbol{P}$.

\n

Find $\\boldsymbol{P}$.

\n

$\\boldsymbol{P} = $ [[0]]

", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": true, "customMarkingAlgorithm": "", "markPerCell": false, "correctAnswer": "p", "allowResize": false, "unitTests": [], "correctAnswerFractions": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "type": "matrix", "numColumns": 1, "tolerance": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": "3", "numRows": "3"}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([c,d,f])", "name": "w", "description": ""}, "s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s1", "description": ""}, "v": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([a,b,g])", "name": "v", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(2..9)", "name": "b", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "ga", "name": "f1", "description": ""}, "s4": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s4", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+lam*c-mu*al", "name": "a1", "description": ""}, "mu": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s2*random(1..5)", "name": "mu", "description": ""}, "be": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "name": "be", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "a", "description": ""}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(a,b,g)+lam*vector(c,d,f)", "name": "p", "description": "

Point of intersection of the two lines

"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(2..9)", "name": "c", "description": ""}, "g1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "g+lam*f-mu*ga", "name": "g1", "description": ""}, "lam": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s3*random(1..5)", "name": "lam", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s4*random(2..9)", "name": "d", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "f", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b+lam*d-mu*be", "name": "b1", "description": ""}, "al": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "name": "al", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "be", "name": "d1", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "al", "name": "c1", "description": ""}, "ga": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "name": "ga", "description": ""}, "s3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s3", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s1*random(2..9)", "name": "g", "description": ""}}, "ungrouped_variables": ["a", "a1", "al", "b", "b1", "be", "c", "c1", "d", "d1", "f", "f1", "g", "g1", "ga", "lam", "mu", "s1", "s2", "s3", "s4", "v", "w", "p"], "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

You are given the vectors

\n

\\begin{align}
\\boldsymbol{x_0} &= \\var{vector(a,b,g)} , & \\boldsymbol{v} & = \\var{vector(c,d,f)}, \\\\[1em]
\\boldsymbol{y_0} &= \\var{vector(a1,b1,g1)}, & \\boldsymbol{w} &=\\var{vector(c1,d1,f1)}
\\end{align}

\n

in $\\mathbb{R^3}$.

", "tags": ["checked2015", "equation of a line", "equation of a line through a vector in the direction of another vector", "Finding a common point for two lines in three dimensional space", "intersection of two lines in three dimensional space", "lines in three dimensional space", "three dimensional geometry", "vector equation of a line", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Given two 3 dim vectors, find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.

"}, "advice": "

a)

\n

\\[\\boldsymbol{r} = \\var{vector(a,b,g)} + \\lambda \\var{vector(c,d,f)}\\]

\n

b)

\n

\\[\\boldsymbol{r} = \\var{vector(a1,b1,g1)} + \\mu \\var{vector(c1,d1,f1)}\\]

\n

c)

\n

Write out a set of simultaneous equations for each component of $\\boldsymbol{P}$:

\n

\\begin{align}
\\simplify[]{{a} + lambda*{c}} &= \\simplify[]{{a1} + mu*{c1}} \\\\
\\simplify[]{{b} + lambda*{d}} &= \\simplify[]{{b1} + mu*{d1}} \\\\
\\simplify[]{{g} + lambda*{f}} &= \\simplify[]{{g1} + mu*{f1}}
\\end{align}

\n

By solving these equations, we find that the point $\\boldsymbol{P}$ common to both lines is given by $\\lambda=\\var{lam},\\mu=\\var{mu}$, and

\n

\\[\\boldsymbol{P} = \\var{p}\\]

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Questions on vector arithmetic and vector operations, including dot and cross product, as well as the vector equations of planes and lines.

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