Math 506:VECTORS
SAMPLE VECTOR QUESTIONS:
1.
Given vectors u and v.
u = AB where A(-2, 3) and B(6, 7)
v = (4, - 4)
What is the scalar product of vectors u and v?


Given u and v two vectors that are not opposite.
2.
Which of the following is FALSE?
3.
A)
   
u •v =v •u
C)
 


2(u + v ) = 2u + 2v
B)


 
2u • 3v = 6v • u
D)




2u + 3v = 3u + 2v


Given u and v represented in the Cartesian plane below.
y
1

v
1
x

u
What is the measure of the angle between these vectors, rounded to the nearest hundredth?
Show all your work.
vector problems & ans.
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Math 506:VECTORS
4.
Given the following prism having a rectangular
base.
E
F
B
A
H
G
D
C
Which vector is equivalent to the resultant of the expression AD + HE + AE ?
5.
A)
DH
C)
FB
B)
BE
D)
BC
Vectors u and v are represented in the Cartesian plane below.
u = AB
where A(3, 4) and B(8, 14)
y
B
v = CD
where C(8, 1) and D(5, -5)
→
u
A•
•C
x
→
v
What is the scalar product of vectors u and v?
vector problems & ans.
D
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Math 506:VECTORS
6.
A hot air balloon is flying due south at 50 km/h.
N
Suddenly, the wind starts blowing from the southeast at
8 km/h.
What is the resulting speed and direction of the hot air
balloon?
N-W
N-E
W
E
S-W
S-E
S
Show all your work
7.
The Egyptians used an ingenious pulley system to move the blocks of stone used in the
construction of pyramids. To minimize the work needed to displace the blocks, they applied a
force oriented at 26°. (Work (Nm) is the scalar product of the force vector and the displacement
vector.)
Rounded to the nearest Nm, what work is needed to displace a block of stone horizontally for a
distance of 200 m, if the force applied to it is 1500 N oriented at 26o?
A)
131 511 Nm
C)
228 768 Nm
B)
194 076 Nm
D)
269 638 Nm
vector problems & ans.
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Math 506:VECTORS
8.
The following figure represents a right prism.
H
E
D
G
F
C
A
B
Which of these statements is FALSE?
A)
BC + GF = 0
C)
AB • AD = 0
B)
AB − FE = 0
D)
EH + HF + FG − EG = 0
9.
Given vector parallelogram PQRS .
Q
P
R
S
Which of the following statements is FALSE?
A)
PQ + QR = PR
B)
RP − SP = RS
C)
PS + SR = RP
D)
SQ + QR + RS = O
vector problems & ans.
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Math 506:VECTORS
10.
An airplane leaves airport A and must fly to
airport B. In the Cartesian plane on the right,
these airports are represented by points A and
B respectively. The scale of the graph is in
kilometres.
N
E
O
y
During the flight, the plane encounters a steady
wind. This wind is represented by the vector v
= (20, -15).
S
B (400, 200)
A (150, 125)
x
The pilot steers the plane so as to negate the effect of the wind.
To the nearest degree, at what angle relative to the east should the pilot point the plane in order
to reach airport B?
Show all your work.
vector problems & ans.
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Math 506:VECTORS
Answers:
1.
The scalar product of vectors u and v is 16.
2.
D
3.

Given u = (5, 2 )
and

v = (2, 3)
Scalar product
 
u • v = (5, 2 ) • (2, 3)
 
u • v = 5× 2 + 2× 3
 
u • v = 16
Magnitude of the vectors

u = 5 2 + 2 2 = 29 ≈ 5.385

v = 2 2 + 3 2 = 13 ≈ 3.606


Angle between u and v
   
u • v = u v cos θ
 
u •v
16
cos θ =   =
u v
29 13
cos θ ≈ 0.824 04
θ ≈ 34.51°
Answer
The angle between these two vectors measures 34.51°.
4. A
5.
The scalar product of vectors u and v is -75.
6.
Given u the resultant vector

vector problems & ans.
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Math 506:VECTORS
Magnitude of the resultant vector
2
u = 82 + 502 − 2 × 8 × 50 × cos 45°
2
u ≈ 1998.31

u ≈ 44.7
θ
u
d
Direction
44.7
8
=
sin 45° sin θ
sin θ = 0.1266
θ = 7.3°
270° - 7.3° = 262.7°
Answer:
7.
D
8.
B
v
The resulting speed of the hot air balloon is 44.7 km/h in a direction of 262.7°.
9. C
10. Components of vector AB
AB = (400 − 150, 200 − 125) = (250, 75)
Components of the unknown vector
v + ? = AB
B
? = AB − v
AB
? = (250, 75) − (20, -15)
? = (250 − 20, 75 + 15)
? = (230, 90)
?
A
v
Direction of the unknown vector
90
tan θ =
230
θ ≈ 21.37°
?
90 km
θ
230 km
vector problems & ans.
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Math 506:VECTORS
Answer:
To the nearest degree, the pilot should point the plane at an angle of 21° relative
to the east in order to reach airport B
vector problems & ans.
Page 8