Unit 5: Vectors
Study Guide
1. What is the magnitude of the vector 6,15
a)
21
b) 261
c) 21
d)
261
2. Give the component form of the vector with head at (3,-8) and the tail at (0,5).
a) 3,8
b) 0,5
c) 3,13
d)  3,13
3. Convert -5 + 6i into polar form




4. Evaluate 2cis  3cis   6cis
4
6
6

5. Give the direction of the vector  8,1
6. A projectile is fired with an initial velocity of 300 ft/sec at an angle of 12º. What
is the vertical component of the velocity?
a) -160.97
b) 253.16
c) 62.37
d) 293.44
7. Find the 3 third roots of unity.
8.
4  7i 6
9. A ship is pulled by two tugboats. One pulls with a force of 8.6 tons at a heading
(direction) of 30º. The second pulls with a force of only 7 tons. What should his
heading be so that the ship is pulled with a heading of 50º?
10. Vector v has tail and head (2,3,4) and (8,12,22) respectively.
Vector u has tail and head (5,2,9) and (9,8,21) respectively.
Are v and u
a. perpendicular?
b. parallel?
c. coincident? (congruent / equivalent)
d. none of the above
11. The projection of vector b onto vector a is given by proj a b 
a b
a
2
a.
a  3,5 b   3,5 . Find proj a b .
12. A three way tug–o–war is set up by tying three heavy ropes to a sturdy metal ring.
The teams set up and, at the whistle, begin pulling. Team a pulls at 30º with a
force of 150lbs. b pulls at 120º with a force of 260lbs. c a pulls at 270º with a
force of 30lbs. What is the net force on the ring (what is the magnitude of the
resultant vector)?
13. Find z so that (2, -4, 3) is orthogonal to (-3, -2, z).
Plot the following complex number s on the polar plane to the right (first convert):
14. 2  2 3i
10
15. 4  4i
8
16. 6
6
17. -4i
4
18.  2  8i
2
2
2  



 i sin
19.  cos
  5 cos  i sin 
3
3  
4
4

3
3 

15 cos
 i sin

2
2 

20.



5 cos  i sin 
4
4

10
5
5
2
4
6
8
5 

21.  2cis

6 

3
10
10
Plot the following complex numbers on the complex plane



22. 2 cos  i sin 
3
3

23.
10
8
6
3 cos   i sin  
4
24.
5cis
3
4
2
-10
 8 81 8 27i 

25. 


2
2


4
5
-5
-2
-4
-6
-8
-10
26. Solve for all solutions of x.
x 5  6cis
27. Find the 4 fourth roots of unity.

8
10
28. The lemniscate shown is given by the equation r( )  15cos(2 ) . Find the area of
the shaded region. For full credit, you must i) identify your t–step () ii) identify –
min and –max for your calculations iii) show a general method of solution (not
necessarily all the calculations but an idea of your reasoning)