Vectors: Unit Vectors 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

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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: Unit Vectors 1.

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

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Question statement

Find the unit vectors in the direction of

$\mathbf a = \var{a}, \quad \mathbf b = \var{b}, \quad \mathbf c = \var{c}, \quad \mathbf d = \simplify{{m}*mathbf:a +{n}*mathbf:b}.$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	vector	vector(-1,3,-5)
b	vector	vector(5,-2,4)
c	vector	vector(2,3,-1)
d	vector	vector(10,9,-13)
m	integer	5
n	integer	3
unitb	vector	vector(0.75,-0.30,0.60)
unita	vector	vector(-0.17,0.51,-0.85)
unitc	vector	vector(0.53,0.80,-0.27)
unitd	vector	vector(0.53,0.48,-0.69)

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(repeat(random(-5..5 except 0),3))

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(-1,3,-5)

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0 Matrix entry
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2. Gap Gap 1 Matrix entry
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3. Gap Gap 2 Matrix entry
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4. Gap Gap 3. Matrix entry
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Gap-fill

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

$\mathbf{\hat{a}}=$ Gap 0 $\quad\mathbf{\hat{b}}=$ Gap 1 $\quad\mathbf{\hat{c}}=$ Gap 2 $\quad\mathbf{\hat{d}}=$ Gap 3.

(Give answers to 2 decimal places where necessary)

Advice

A unit vector in the direction of a vector $\mathbf n$ is defined as

$\mathbf{\hat{n}} = \dfrac{\mathbf{n}}{|\mathbf n |} ,$

where $|\mathbf n |$ is the magnitude of $\mathbf n$ .

For the vector $\mathbf a = \var{a}$ ,

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

Therefore,

$\mathbf{\hat{a}} = \frac{1}{\simplify{sqrt({length(a)^2})}}\var{a} = \var{unita}.$

Similarly,

$\mathbf{\hat{b}} = \frac{1}{\simplify{sqrt({length(b)^2})}}\var{b} = \var{unitb},$

and

$\mathbf{\hat{c}} = \frac{1}{\simplify{sqrt({length(c)^2})}}\var{c} = \var{unitc}.$

For the vector $\mathbf d = \simplify{{m}*mathbf:a +{n}*mathbf:b}$ :

$\mathbf d = \simplify[!collectNumbers]{{m}*{a} +{n}*{b}} = \var{d}.$

Therefore,

$\mathbf{\hat{d}} = \frac{1}{\simplify{sqrt({length(d)^2})}}\var{d} = \var{unitd}.$

Note: It is also acceptable to give the answer in the form

$\mathbf{\hat{n}} = \dfrac{1}{|\mathbf n |}\mathbf{n},$

rather than dividing each term by the magnitude.

Give a worked solution to the whole question.

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This question is used in the following exams:

Vectors by Ben McGovern in MASH Bath: Moodle quizzes and TS.
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.
Vector by Dana Mackey in Dana's workspace.

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Taxonomy: mathcentre

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Taxonomy: Context

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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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Generated value: `vector`

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Gap-fill

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Vectors: Unit Vectors 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`