Multivariable Calculus
Exam 1
Name:_____________________________________________________
possible points
1.
25
2.
25
3.
25
4.
25
5.
25
6.
25
7.
25
8.
25
9.
25
total possible points =
225
earned points
You must show all your work. The number of points earned on each problem
will depend upon how well you have justified your solution.
1. The distance between the two points A(1,-5,a) and B(10,-2,4a) is 6 15 units. What is
the value of a?
6 15  92  32  9a 2
540  90  9a 2
450  9a 2
a 2  50
a5 2
2. Vectors a and b, as shown below, have magnitudes 6 and 8 respectively. What is the
sum of a and b?
a
30o
60o
b
a  6 cos 30 i  6sin 30o j  3 3i  3 j
b  8cos 60 i  8sin 60o j  4i  4 3
a  b  (3 3  4)i  (3  4 3) j  1.196i  3.9282
3. Find three distinct unit vectors orthogonal to 1,3, 7 .
There are an infinite number of possibilities. Three such are:
3,1,0 7,0,1 0,7,3
,
,
10
5 2
58
4. Calculate the following when give a  i  j  k and b  3i  2 j and c  j  7k .
i) a  b
ii)  a b    a c 
i
j
k
i) a  b  1 1 1  2i  3j  5k
3
2
0
ii)  a b    a c   1  8  7
5. Determine if the following vectors are coplanar.
a  1,0, 2 , b  0,5, 3 , c  1,1,1
Find the volume of a parallelepiped determined by the three vectors.
1 0 2
0 5 3  1(8)  2(5)  18 . Since the parallelepiped has volume, the vectors are not
1 1 1
coplanar.
6. The line l passes through the two points P(1,2,3) and Q(-1,0,4). Find parametric
equations for the line that is parallel to l containing the point (0,1,-2).
PQ  2, 2,1 the direction of l.
x  0  2t
y  1  2t
z  2  t
7. Determine whether the planes are parallel. If they are not parallel determine the angle
between them.
3y  2z  8
5 x  2 y  z  10
Normal vectors of the planes are 0,3, 2 and 5,-2,1 . Since one is not the scalar
multiple of the other, they are not parallel. The angle between them is
062
8
arccos(
)  arccos(
)  113.9
9  4 25  4  1
13 30
8. Find the scalar and vector projections of v onto w when v  2,1,3 , w  5,0, 1 .
10  0  3
7
.

25  0  1
26
35,0, 7
7 5,0, 1
The vector projection is

26
26
26
The scalar projection is
9. Find a set of traces ( at least three) on the xy-plane of the quadric surface
x 2  9 y 2  z 2  81 .