Vectors: Vector Product 1

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Name	Status	Author	Last Modified
Vectors: Vector Product 1	Ready to use	Ben McGovern	10/10/2022 13:34
Vectors: Vector Product 2	Ready to use	Ben McGovern	10/10/2022 13:34
Vectors: Vector Product 3	Ready to use	Ben McGovern	10/10/2022 13:34

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Question statement

Given the vectors $\mathbf a = \var{a}$ and $\mathbf b = \var{b}$ , calculate $\mathbf a \, \times \mathbf b$ .

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			Name	Type	Generated Value

a	vector	vector(3,1,5)
b	vector	vector(3,-3,3)
c	vector	vector(18,6,-12)

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(random(-2..6),random(-4..4 except 0),random(1..7))

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Description

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Generated value: `vector`

vector(3,1,5)

This variable doesn't seem to be used anywhere.

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1. Gap Matrix entry
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$\mathbf a \, \times \mathbf b =$ Unnamed gap

Advice

To find the vector product of the vectors $\mathbf a$ and $\mathbf b$ , we can use the formula

$\simplify{cross(mathbf:a,mathbf:b)} = (a_2 b_3 - a_3 b_2)\mathbf i - (a_1 b_3 - a_3 b_1) \mathbf j + (a_1 b_2 - a_2 b_1)\mathbf k.$

NOTE: This is equivalent to finding the determinant of the matrix

$\begin{pmatrix} \mathbf i &\mathbf j &\mathbf k \\ a_1 &a_2 &a_3 \\ b_1 &b_2 &b_3 \end{pmatrix}$

So, for the vectors $\mathbf a = \var{a}$ and $\mathbf b = \var{b}$ ,

$\begin{split}\simplify{cross(mathbf:a,mathbf:b)} &\,= \begin{vmatrix} \mathbf i &\mathbf j &\mathbf k \\ \var{a[0]} &\var{a[1]} &\var{a[2]} \\ \var{b[0]} &\var{b[1]} &\var{b[2]} \end{vmatrix}\\ \\&\,= (\simplify[!collectNumbers]{{a[1]}*{b[2]}-{a[2]}*{b[1]}}) \mathbf i - (\simplify[!collectNumbers]{{a[0]}*{b[2]}-{a[2]}*{b[0]}}) \mathbf j + (\simplify[!collectNumbers]{{a[0]}*{b[1]}-{a[1]}*{b[0]}}) \mathbf k \\\\ &\,= \simplify[all,!noLeadingMinus]{{c[0]}*mathbf:i+ {c[1]}*mathbf:j + {c[2]}*mathbf:k}\end{split}$

Therefore,

$\simplify{cross(mathbf:a,mathbf:b) = {c}} .$

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This question is used in the following exams:

Vectors - Scalar Product and Vector Product by Ben McGovern in MASH Bath: Moodle quizzes and TS.
Training Space: Vectors by Ben McGovern in MASH Bath: Moodle quizzes and TS.
Vectors by Chris Templet in Chris's workspace.
Vector Practice by Rabail Tahir in Rabail's workspace.
Vectors by Dean Ang in Dean's workspace.
Vector Exam by Dean Ang in Dean's workspace.
MASH question trig and vectors by Morgiane Richard in Mathematics for Geosciences.
Mathematical Sciences Training Space by Ben McGovern in MASH Bath: Moodle quizzes and TS.

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Christian Lawson-Perfect 2 years, 7 months ago

Ben McGovern 4 years, 1 month ago

Ben McGovern 4 years, 1 month ago

Error in variable testing condition

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Generated value: `vector`

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Vectors: Vector Product 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 2 years, 7 months ago

Ben McGovern 4 years, 1 month ago

Ben McGovern 4 years, 1 month ago

Error in variable testing condition

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Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`