Name: _______________________________
Vectors Practice
1. A particle is moving in three dimensions. Its position vector is given by:

r  6 î  (3  4t) ĵ  (3  2t  t 2 ) k̂
Distances are in meters and time is in seconds.

A. What is the magnitude of r at t = 0?
B. What is the velocity vector at t = +3?
C. What is the speed at t = +3?
D. What is the acceleration vector and what is its magnitude at t = +3?
Now the particle is moving along the z axis and its position is given by:

r  (t 2  2t  3) k̂
E. At what time does the particle stand still?
F. Sketch a graph of z versus time covering t = -2 to +4 seconds.
Vector Magnitudes, Direction, Addition and Subtraction
2. For the vectors


a  3.0 î  4.0 ĵ m
b  5.0 î  2.0 ĵ m

A. What is the magnitude of a and what angle does it make with respect to î (i.e. the
positive x-axis)?

B. What is the magnitude of b and what angle does it make with respect to î ?

  
C. Find c where c  a  b .
 

D. Sketch each individual vectors a , b and c.

  
E. Find d where d  a  b .

 
F. Sketch each individual vectors a , b and d.
Vector Components

3. A projectile is launched with an initial velocity, v0 , of 75 m/s at an angle of 62o with
respect to the x-axis.

A. What is the x-component of v0 ?

B. What is the y-component of v0 ?

C. Write v0 in unit vector notation.
4. Three forces are applied to a particle.
F1 = 14.4 N at 90.0o
F2 = 20.2 N at 135o
F3 = 10.5 N at -165o
Find the resultant force vector (magnitude & angle and written in unit vector notation).
[Hint: Fill in the table and add the components.]
Fx
Fy
F1
F2
F3
Total
Dot Product/Scalar Product
5. For the vectors

a  3.0 î  4.0 ĵ m
 
A. Find S where S  a b using components.

b  5.0 î  2.0 ĵ m


B. Use the definition of the dot/scalar product to find the angle between a and b. Is this
consistent with the angles you found in 2A and 2B above?
6. The following two vectors are perpendicular to each other.


g  6.0 î  2.0 ĵ m
h  x î  9.0 ĵ m
What is the value for x?
7. A 7.50 N force is applied an object at an angle of 60.0o. The object’s displacement is 15.0
m in a direction of 20.0o with respect to the x-axis. What is the work done in moving this
 
object? [Note: W  F  d ]
Cross Product/Vector Product
8. For the vectors

v  2 î  3 ĵ  4 k̂

w  5 î  ĵ  2 k̂
 
A. Write v x w as a matrix.
 
B. Solve the matrix and write v x w unit vector notation.
9. A charge q = 3 coulombs (C) is traveling in the x-y plane at 5 m/s at an angle of 45o with

respect to the x-axis in a magnetic field given by B  10 î tesla (T). Find the magnitude and

 
direction of the force on this charge. [Note: F  qv x B ]
10. Two vectors are perpendicular to one another. One has a magnitude of 25. The other has
a magnitude of 4. What is the magnitude of their cross product?
11. Two vectors are separated by an angle of 120o. One has a magnitude of 12.0. The cross
product of these two vectors has a magnitude of 64.0. What is the magnitude of the
second vector?
Solutions:
1. A. 3 6 m (= 7.35 m)
2. A. 5 m, 53o
3. A. 35 m/s
B. 4 ĵ  4k̂
B. 5.4 m, -22o
B. 66 m/s
C. 8.0î  2.0 ĵ
C. 35î  66ĵ m / s
4. 35.6 N @ 133o,  24.4î  26.0 ĵ
5. A. 7.0 m2
B. Yes
6. 3.0 m
7. 86.2 J
î
ĵ k̂
8. A. 2 3 4
5 1 2
B. 10î  16ĵ  17k̂
9. 110 N in the -z direction, - 110 k̂ N
10. 100
11. 6.16
C. 4 2
m
(= 5.66 m/s)
s
D. 2k̂
E.  2.0î  6.0 ĵ
m
s
2
,2
m
s2
E. 1 s