Vectors: Scalar Product 1

Error

There was an error loading the page.

Contributors

Ben McGovern

Feedback

From users who are members of MASH Bath: Question Bank :

Ben McGovern

said

Ready to use

4 years, 1 month ago

History

○

Christian Lawson-Perfect 2 years, 7 months ago

Published this.

○

Ben McGovern 4 years, 1 month ago

Gave some feedback: Ready to use

○

Ben McGovern 4 years, 2 months ago

Created this.

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

There are 22 other versions that do you not have access to.

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(4,3)
b	vector	vector(2,3)
c	vector	vector(3,4)
adotb	number	17
angleab	number	19.44
abdotc	vector	vector(72,54)

Automatically regenerate variables?

Name

Data type

Value

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

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(4,3)

This variable doesn't seem to be used anywhere.

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 $\mathbf v = \pmatrix{v_1 \\ v_2},\, \mathbf w = \pmatrix{w_1 \\ w_2},$

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

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 $\mathbf b = \var{b}$ ,

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

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

and

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

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 $\mathbf c$ :

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

Therefore,

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

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
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet

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.
Week 18 e-quiz by Ruth Douglas in Ruth's workspace.
Vectors by Dean Ang in Dean's workspace.
My First Exam by George Li in George'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.

Loading...

Error

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, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `vector`

Parts

Gap-fill

Loading...

Error

Vectors: Scalar 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, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`