Scalar triple product of coplanar vectors

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Bill Foster

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6 years, 4 months ago

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Bill Foster 6 years, 4 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Scalar triple product of coplanar vectors	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Scalar Triple Product	draft	Matthew Mears	27/03/2020 21:27

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Question statement

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

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

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

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

Give any introductory information the student needs.

			Name	Type	Generated Value

w1	integer	12
w2	integer	-4
w3	integer	44

			Name	Type	Generated Value

s1	integer	1
s2	integer	1
s3	integer	-1
s4	integer	1
s5	integer	-1
s6	integer	1

			Name	Type	Generated Value

x1	integer	8
x2	integer	2
x3	integer	-2

			Name	Type	Generated Value

y1	integer	-6
y2	integer	4
y3	integer	2

			Name	Type	Generated Value

			z1	integer	-7
			z2	integer	4

			Name	Type	Generated Value

			ty3	integer	2
			lambda	rational	25/11

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x2*y3-y2*x3

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Generated value: `integer`

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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 $\boldsymbol{b}$ .

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.

Now

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

Hence

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

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

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

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This question is used in the following exams:

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Bill Foster 6 years, 4 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Scalar triple product of coplanar vectors

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 4 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`