Vectors: Scalar Product 2

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Ben McGovern

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Ben McGovern

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4 years, 1 month ago

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Christian Lawson-Perfect 2 years, 7 months ago

Published this.

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Ben McGovern 4 years, 1 month ago

Gave some feedback: Ready to use

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Ben McGovern 4 years, 1 month ago

Created this as a copy of Vectors: Scalar Product 1.

Name	Status	Author	Last Modified
Vectors: Scalar Product 1	Ready to use	Ben McGovern	10/10/2022 13:33
Vectors: Scalar Product 2	Ready to use	Ben McGovern	10/10/2022 13:33

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Question statement

Given the vectors

$\mathbf a = \var{a},\quad \mathbf b = \var{b}, \quad \mathbf c = \var{c};$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	vector	vector(4,-4,3)
b	vector	vector(-1,4,5)
c	vector	vector(-4,5,-6)
adotb	number	-5
angleab	number	96.92
abdotc	vector	vector(-24,24,-18)

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(random(1..7),random(-4..4 except [0,1]),random(-3..5))

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: `vector`

vector(4,-4,3)

→ Used by:

abdotc
adotb
angleab
c

Statement
Advice

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) Gap-fill
1. Gap Mathematical expression
  Add
Part c) Gap-fill
1. Gap Matrix entry
  Add

Gap-fill

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

Calculate the scalar product between $\mathbf a$ and $\mathbf b$ :

$\mathbf{a \cdot b} =$ Unnamed gap

Advice

To answer these questions, we want to use the equations for the scalar product. Recall:

For the vectors $\mathbf v = \pmatrix{v_1 \\ v_2 \\ v_3},\, \mathbf w = \pmatrix{w_1 \\ w_2 \\ w_3},$

$\begin{split} \mathbf{v \cdot w} &\,= v_1 \times w_1 + v_2 \times w_2 + v_3 \times w_3 \\\\ \mathbf{v \cdot w} &\,= |\mathbf v| |\mathbf w | \cos(\theta), \end{split}$

where $|\mathbf v|$ and $|\mathbf w|$ are the magnitudes of the vectors, and $\theta$ is the angle between the vectors.

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

$\begin{split} |\mathbf a| &\,= \simplify[!collectNumbers]{sqrt({a[0]}^2+{a[1]}^2+{a[2]}^2)} \\ &\,=\var{precround(length(a),2)}, \end{split}$

$\begin{split} |\mathbf b| &\,= \simplify[!collectNumbers]{sqrt({b[0]}^2+{b[1]}^2+{b[2]}^2)} \\ &\,=\var{precround(length(b),2)}, \end{split}$

and

$\begin{split} \mathbf{a \cdot b} &\,= \simplify[alwaysTimes,!collectNumbers]{{a[0]}*{b[0]}+{a[1]}*{b[1]}+{a[2]}*{b[2]}} \\ &\,= \var{adotb}. \end{split}$

Using these results to find the angle between $\mathbf a$ and $\mathbf b$ :

$\begin{split} \cos(\theta) &\,= \frac{\mathbf{a \cdot b}}{|\mathbf a| |\mathbf b |} , \\ \\ &\,=\frac{\var{adotb}}{\var{precround(length(a),2)}\times\var{precround(length(b),2)}} \\\\ &\,=\var{precround(adotb/(length(a)*length(b)),2)}.\end{split}$

Therefore,

$\begin{split} \theta &\,= \cos^{-1}(\var{precround(adotb/(length(a)*length(b)),2)}) \\ &\,= \var{angleab}^\circ . \end{split}$

To calculate $\mathbf a \left( \mathbf{b \cdot c} \right)$ , we want to first find the dot product between $\mathbf b$ and $\mathbf c$ :

$\begin{split} \mathbf{b \cdot c} &\,= \simplify[alwaysTimes,!collectNumbers]{{b[0]}*{c[0]}+{b[1]}*{c[1]}+{b[2]}*{c[2]}} \\ &\,= \var{dot(b,c)}. \end{split}$

Therefore,

$\begin{split} \mathbf a \left( \mathbf{b \cdot c} \right) &\,= \var{dot(b,c)} \var{a} \\\\ &\,= \var{abdotc}. \end{split}$

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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.
NCM4 FE-F2022 UNIT 8 All by Terry Young in NCM4 FINAL EXAM Fall 2022.
Vectors by Chris Templet in Chris'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.

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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: Scalar Product 2

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

Error

Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`