Scalar Triple Product

Error

There was an error loading the page.

Contributors

Matthew Mears

History

○

Matthew Mears 7 years, 6 months ago

Created this as a copy of Scalar triple product of coplanar vectors.

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

There are 4 other versions that do you not have access to.

Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

w1	integer	-88
w2	integer	58
w3	integer	42

			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	-1
x2	integer	5
x3	integer	-9

			Name	Type	Generated Value

y1	integer	-7
y2	integer	-7
y3	integer	-5

			Name	Type	Generated Value

			z1	integer	-3
			z2	integer	-7

			Name	Type	Generated Value

			ty3	integer	-5
			lambda	rational	71/21

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
x2*y3-y2*x3

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

-88

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

You are given three vectors$,

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

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.

$\lambda=$ Unnamed gap.

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

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 $(\boldsymbol{a}\times \boldsymbol{b})\boldsymbol{\cdot} \boldsymbol{c}=0$ .

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

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet

This question is not used in any exams.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Matthew Mears 7 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Scalar Triple Product

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Matthew Mears 7 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`