Vectors: Unit Vectors 1

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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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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(-3,-5)
b	vector	vector(5,-3)
c	vector	vector(1,-5)
d	vector	vector(-37,-5)
m	integer	4
n	integer	-5
unitb	vector	vector(0.86,-0.51)
unita	vector	vector(-0.51,-0.86)
unitc	vector	vector(0.20,-0.98)
unitd	vector	vector(-0.99,-0.13)

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector(repeat(random(-5..5 except 0),2))

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(-3,-5)

→ Used by:

b
c
d
unita

This variable doesn't seem to be used anywhere.

Parts

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

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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Taxonomy: mathcentre

Taxonomy: Kind of activity

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, 2 months ago

Error in variable testing condition

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Generated value: `vector`

Parts

Gap-fill

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Vectors: Unit Vectors 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`