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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
please try to submit your calculations as a single file document (preferably as a
Important
When answering questions in a mathematics course always be sure to use the
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
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
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03/08/2022, 00:36
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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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