Prof. Israel N Nwaguru
MATH 2415
CHAPTER 11 - REVIEW.
PLEASE, DO NOT WRITE OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE
PAPER/ PAPERS THEN CHOSE THE BEST ANSWER. AGAIN " NO WORK NO CREDIT".
1. Find the vector v whose initial and terminal points are given below.
(2.2,4.9), (2.125,3.65)
Ans: v  – 0.075i – 1.25 j
2. Find (a) 10u (b) v-u (c) 5u+3v given the following values for u and v .
u  4,6 ,
(a) 10u
Ans: 40,60
v  5,5
(b) v - u
1, –1
3. The vector v and its initial point is given. Find the terminal point.
v  2,2 ,
Ans:  2, 4 
initial point
 0, 2 
4. Find the magnitude of the vector given below.
v  3,1
Ans:
10 or 3.1623
5. Find the unit vector in the direction of u.
u  5, –3
The possible solutions are given to two decimal places.
Ans:
5
–3
,
34 34
6. Given the vectors
u  5,3 ,
v  4, –5
find the following:
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(c) 5u+3v
35,45
Prof. Israel N Nwaguru
MATH 2415
CHAPTER 14 - EXAM 3. PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/ PAPERS THEN
CHOSE THE BEST ANSWER. AGAIN " NO WORK NO CREDIT".
(a) u + v
Ans:
(b)
u
u
(c)
uv
uv
1
85 or 9.22
1
7. Find the component form of vector v given its magnitude and the angle it makes with
the positive x-axis.
v  14,   170
Ans: v  14cos(170),14sin(170)
8. Find the component form of a vector v given the magnitude of u and u+v and the
angles that u and u+v make with the positive x-axis.
u  5,   140 , u  v  9,   20
Ans: v  9cos(20)  5cos(140),9sin(20)  5sin(140)
9. Find the distance between the points given below.
(–5 ,–2, 3 ),
( –3 ,3 , 10 )
Ans:
78 or 8.8318 to four decimal places.
10. Find the coordinates of the midpoint of the line segment joining the points given below.
(–1 ,6, –5 ),
( 1 ,9 , 5 )
Ans: ( 0 ,7.5 , 0 )
11. Find the standard equation of the sphere with center (–4 , 1, –1 ), and radius 5 .
Ans: ( x + 4)2  ( y – 1)2  ( z +1)2  25
12. Complete the square to write the following equation in the standard equation of a
sphere.
x 2  y 2  z 2 +10 x + 6 y – 2 z  10  0
Ans: ( x + 5)2  ( y + 3)2  ( z – 1)2  25
13. Determine whether u and v are orthogonal, parallel, or neither.
u  –16, –4 ,
Ans: Orthogonal
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v  1, –4
Prof. Israel N Nwaguru
MATH 2415
CHAPTER 14 - EXAM 3. PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/ PAPERS THEN
CHOSE THE BEST ANSWER. AGAIN " NO WORK NO CREDIT".
14. Determine whether u and v are orthogonal, parallel, or neither.
u  2i  2 j, v  3i +10j
Ans: Neither parallel nor orthogonal
15. Find the angle between the vectors for u and v.
u  5i  3j,
Ans: 97.70 degrees
v  –4i  5j
16. Given the vectors u nd v, find the cross product and determine whether it is orthogonal
to both u and v.
u  3,8,2 , v  –6,8,7
u v
u  v  u
u  v  v
Ans:
40, – 33, 72
0
0
Orthogonal
17. Find the area of a parallelogram that has the given vectors as adjacent sides.
u  0,5,3 , v  –6, –3,6
Ans:
2745 or 52.39 to two decimal places.
18. Find the triple scalar product of the vectors
u  8,8,4 , v  –1,3, –8 , w  –8, –4,1
Ans:
400
19. Use the triple scalar product to find the volume of the parallelepiped having adjacent
edges given by the vectors
u  3,8,2 , v  –2,8, –8 , w  –6, –7, –7
Ans:
60
20. Find a set of parametric equations of the line through the point  5,9, 4  parallel to the
vector v=(7, 4,2).
Ans: x  5  7s, y  9  4s, z  4  2s
21. Convert the following point from cylindrical coordinates to rectangular coordinates.
Page 3
Prof. Israel N Nwaguru
MATH 2415
CHAPTER 14 - EXAM 3. PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/ PAPERS THEN
CHOSE THE BEST ANSWER. AGAIN " NO WORK NO CREDIT".
10 

, 6
 18,
3


Ans: –9, –9 3,6


22. Convert the following point from rectangular coordinates to cylindrical coordinates.
Give any angles in radians.
8,8,6
Ans: 


 8 2, , 6 
4


23. Find an equation in cylindrical coordinates for the equation given in rectangular
coordinates.
z  49 x 2  49 y 2  2
Ans: None of the above
24. Find an equation in rectangular coordinates for the equation given in cylindrical
coordinates.
r  6sin 
Ans: x 2  y 2  6 y
25. Convert the point from spherical coordinates to rectangular coordinates.
 5 5 
 11, , 
3 6 

Ans:  11 11 3 11 3 
,–
 ,–

4
2 
4
26. Find an equation in spherical coordinates for the equation given in rectangular
coordinates.
y2
Ans:   2csc  csc
27. Find an equation in rectangular coordinates for the equation given in spherical
coordinates.
Page 4
Prof. Israel N Nwaguru
MATH 2415
CHAPTER 14 - EXAM 3. PLEASE, DO NOT WRITE
OR MARK ON THIS EXAM PAPER. SHOW ALL STEP BY STEP ON SEPARATE PAPER/ PAPERS THEN
CHOSE THE BEST ANSWER. AGAIN " NO WORK NO CREDIT".

π
12
The coefficients below are given to two decimal places.
Ans: y  0.27 x
28. Convert the following point from cylindrical coordinates to spherical coordinates.
π


 3 3, , –3 
3


Ans:  π 2 
 6, ,

 3 3 
29. Convert the following point from spherical coordinates to cylindrical coordinates.
 π 5π 
 7, , 
 3 6 
Ans:  7 π –7 3 
 , ,

2 3 2 
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