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Vector Applications Unit Assignment

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Vector Applications Unit Assignment

You may choose to complete your mathematical calculations by hand and scan

or take images to upload to the dropbox. To help your teacher review your work,

please try to submit your calculations as a single file document (preferably as a

pdf). If you arenâ€™t sure how to do this, please contact your teacher for assistance.

Important

When answering questions in a mathematics course always be sure to use the

following guidelines to help you do your best:

ï„‘

Provide full solutions, showing all of your steps.

Make sure that there is one step or idea per line.

ï„‘ Use one equal sign per line.

ï„‘ Make sure that equal signs line up vertically.

ï„‘ Donâ€™t use self-developed short form notations.

ï„‘

1. Given

, determine: (4 marks)

and

a.

b. A unit vector in the direction of

c. The angle between

and

d. A vector perpendicular to

2. A force

.

.

in Newtons, pulls a sled through a displacement

in metres. The link between the dot product and geometric

vectors and the calculation of work is

. (2 marks)

a. How much work is done on the sled by the force?

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Vector Applications Unit Assignment

b. What is the minimum magnitude of force that could have been applied to

the sled to obtain the same work? Explain your answer.

3. Given

a.

, determine: (4 marks)

and

and verify that it is perpendicular to both

b. A vector

such that

between the vectors

4. Given

a.

and

.

. What is the relationship

,

, and

in this case, and why? Verify this.

, determine: (4 marks)

and

b.

c. What does the magnitude of

depend on?

d. What does the direction of

depend on?

5. Draw diagrams to explain the answers to the following questions. (4 marks)

a. Is it possible to have

?

b. Is it possible to have

undefined?

c. Is it possible to have

d. Explain why

?

.

6. Answer the following with either an explanation, a diagram or a proof: (4

marks)

a. If

b. If

7. Prove that

, what is the relationship between

and

, what is the relationship between

for all

,

and

,

?

?

. (2

marks)

8. Given vectors , ,

, and

, state whether each of the following results

in a scalar, a vector, or is not possible. Justify each response. (6 marks)

a.

b.

c.

d.

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Vector Applications Unit Assignment

e.

f.

9. Charlie is trying to hold on to his toy fire truck. His brother Noah is pulling with

a force of 8 N on a bearing of 023Â° and his brother Jude with a force of 5 N on

a bearing of 155Â°. What force does Charlie need to exert to keep the toy in

equilibrium? (6 marks)

10. A pilot wishes to fly from Bayfield to Kitchener, a distance of 100 km on a

bearing of 105Â°. The speed of the plane in still air is 240 km/h. A 20 km/h wind

is blowing on a bearing of 210Â°.

Remembering that she must fly on a bearing of 105Â° relative to the ground

(i.e. the resultant must be on that bearing), determine: (6 marks)

a. the heading she should take to reach her destination.

b. how long the trip will take.

Submit this assignment to the dropbox. This assignment will be evaluated for a grade that will

contribute to your overall final grade in this course.

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Projection

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Projection

â‡€

When solving for projections, the proper notation for the projection of â‡€

u onto v

is:

â‡€

â‡€

u â†“ v

Definition

â‡€

â‡€

â‡€

The projection of â‡€

u onto v is obtained by moving the tail of u onto v and

âˆ’

â‡€

â‡€

â‡€

u

OB

u

dropping a perpendicular line from the head of . The

is the projection of

â‡€

â‡€

â‡€

â‡€

on â‡€

v . In other words, u â†“ v is just the shadow of u on v . The projection is

also known as orthogonal projection.

Formula Derivation

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Projection

âˆ’

â‡€

â‡€

OB

v

If you look at the above diagram, vector

is a scalar multiple of the vector

.

â‡€

âˆ’

â‡€

OB

v

In other words,

and

has the same direction, but different magnitude.

Let

â‡€

âˆ’

â‡€

OB = s v

Assumption:

âˆ˜

0 â‰¤ Î¸ â‰¤ 90

âˆ’

âˆ’

â‡€

â‡€

â‡€

OB = s v

OB

Since

of â‡€

v .

is equal to s multiplied by the magnitude

, the magnitude of

That is,

â‡€

âˆ’

â‡€

OB = s v

âˆ’ âˆ’ âˆ’ âˆ’(1)

In Î”OAB,

â‡€

OB

cos(Î¸)

=

â‡€

u

â‡€

âˆ’

â‡€

OB

=

u cos(Î¸) âˆ’ âˆ’ âˆ’ âˆ’(2)

But, we know that:

â‡€ â‡€

u â‹… v

cos(Î¸) =

â‡€

u

â‡€

v

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âˆ£âˆ£

Projection

Therefore,

â‡€

âˆ’

â‡€

u â‹…â‡€

v

â‡€

OB

=

u

â‡€

u

â‡€

u â‹…â‡€

v

=

â‡€

v

â‡€

v

âˆ’ âˆ’ âˆ’ âˆ’(3)

Compare equation (1) and equation (3):

s

Therefore,

â‡€

v

s

â‡€ â‡€

u â‹… v

=

=

=

â‡€

v

â‡€ â‡€

u â‹… v

â‡€

v

â‡€

v

â‡€ â‡€

u â‹… v

â‡€

v

2

â‡€

Substitute s into equation (1) to get the projection of â‡€

u on v .

â‡€ â‡€

â‡€

âˆ’

u â‹… v

â‡€

OB =

v

2

â‡€

v

That is,

â‡€

u

â‡€

v =

â†“

â‡€ â‡€

u â‹… v

â‡€

v

2

â‡€

v , where

â‡€

v

â‰ 0

Scalar Projection

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Projection

â‡€

â‡€

â‡€

The scalar projection of â‡€

u on v is the magnitude of u â†“ v .

That is,

â‡€ â‡€

u â‹… v

â‡€

v

Vector Projection

â‡€

u

â‡€ â‡€

u â‹… v

â‡€

v =

â‡€

v

â†“

2

â‡€

v

Example

ïŠ˜

â‡€

â‡€

â‡€

If â‡€

u = [2, âˆ’3] and v = [6,2]. Determine the projection of u on v .

Solution

â‡€

â‡€

u â†“ v

=

=

â‡€ â‡€

u â‹… v

2

â‡€

v

â‡€

v

(2)(6)+(âˆ’3)(2)

2

[6,2]

âˆš 62+22

=

=

=

=

12âˆ’6

36+4

6

40

3

20

[

[6,2]

[6,2]

[6,2]

9

10

,

3

10

]

Quiz

â‡€

If â‡€

u = [3,1] and v = [-4, 3], answer the following questions.

â‡€

Select the notation and solution of the projection of â‡€

u on v .

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â

Projection

[1.44, -1.08]

â†’

â†’

v â†“ u

[-2.70, -0.90]

â†’

â†’

u â†“ v

â‡€

Select the notation and solution of the projection of â‡€

v on u .

[1.44, -1.08]

â†’

â†’

u â†“ v

[-2.70, -0.90]

â†’

â†’

v â†“ u

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