aba = 5, 1, 6 b = 1, 0, 3 ab = 3, 21, 1 ab = 6, 5, 1 ab = 21, 1, 3 ab = 1 ...

Name: __________________
1 Find the cross product a b .
a = 5, 1, 6 , b = 1, 0, 3
Class: Math 22
Spring 2005
Date: _____________
4 Find two unit vectors orthogonal to both 4, 1, 2 and
2, 3, 8 .
a. a b = 3, 21, 1
b. a b = 6, 5, 1
c. a b = 21, 1, 3
d. a b = 1, 1, 9
e. a b = 3, 21, 1
2 Find the cross product a b .
a = t , t 2 , t 3 , b = 1, 5t , 8t 2
a. a b = 3t 3 i 7t 2 j + 4t k
b. a b = 3t 4 i 7t 3 j 4t 2 k
a.
b.
c.
2
,
1
,
1
6 6 6
2
6
,
1
,
6
1
,
6
1
,
2
,
6 6
1
,
6
1
,
6
1
6
2
,
1
6 6
1
6
1
,
2
6 6
1
,
2
6 6
,
,
,
c.
d.
e.
a b = 3t 2 i 7t 2 j + 4t 2 k
a b = 3t 4 i + 7t 3 j 4t 2 k
a b = 3t 4 i 7t 3 j + 4t 2 k
d.
1
,
6
1
,
6
2
,
6
2
,
6
1
6
1
6
,
3 The figure shows a vector a in the xy plane and a
vector b in the direction of k . Their lengths are a =
6 and b = 2 . Find a b .
e.
1
,
6
1
,
1
,
6 6
1
,
2
6 6
2
6
,
5 Find the area of the parallelogram with vertices
K ( 4, 6, 2) , L ( 8, 8, 8) , M ( 13, 10, 13)
, and
N ( 9, 8, 7)
. Please round the answer to the nearest
hundredth.
________
a b =
________
6 Find a vector orthogonal to the plane through the points P,
Q, and R.
P ( 3, 0, 0) , Q ( 0, 9, 0) , R ( 0, 0, 4)
a. 36, 11, 31 d. 34, 11, 32
b. 36, 12, 27 e. 35, 5, 31
c.
35, 11, 27
PAGE 1

Name: __________________
7 Find a vector equation and parametric equations for the line
through the point ( 6, 0, 10)
and parallel to the vector
10i 10j + 6k .
a.
b.
c.
d.
e.
r = ( 4 + 13t ) i 13t j + ( 8 + 9t ) k;
x =
4 + 13t , y = 13t , z = 8 + 9t
r = ( 9 + 14t ) i 14t j + ( 13 + 10t ) k ;
x = 9 + 14t , y = 14t , z = 13 + 10t
r = ( 7 + 8t ) i 8t j + ( 11 + 4t ) k;
x =
7 + 8t , y = 8t , z = 11 + 4t
r = ( 10 + 11t ) i 11t j + ( 14 + 7t ) k;
x = 10 + 11t , y = 11t , z = 14 + 7t
r = ( 6 + 10t ) i 10t j + ( 10 + 6t ) k;
x =
6 + 10t , y = 10t , z = 10 + 6t
8 Find a vector equation and parametric equations for the line
through the origin and parallel to the line x = 8t , y =
7 9t , z = 9 + 8t .
a.
b.
c.
d.
e.
r = 7t i 8t j + 7t k;
x = 7t , y = 8t , z =
7t
r = 8t i 9t j + 8t k;
x = 8t , y = 9t , z =
8t
r = 12t i 13t j + 12t k ; x = 12t , y =
13t , z = 12t
r = 5t i 6t j + 5t k;
x = 5t , y = 6t , z =
5t
r = 6t i 7t j + 6t k;
x = 6t , y = 7t , z =
6t
9 Find a vector equation and parametric equations for the line
through the point ( 7, 0, 10)
and perpendicular to the
plane 9x + 7y + 8z = 8.
a.
b.
c.
d.
e.
Class: Math 22
Spring 2005
r = ( 6 + 11t ) i + 9t j + ( 9 + 10t ) k;
x = 6 +
11t , y = 9t , z = 9 + 10t
r = ( 9 + 12t ) i + 10t j + ( 12 + 11t ) k ; x =
9 + 12t , y = 10t , z = 12 + 11t
r = ( 10 + 13t ) i + 11t j + ( 13 + 12t ) k ; x =
10 + 13t , y = 11t , z = 13 + 12t
r = ( 7 + 9t ) i + 7t j + ( 10 + 8t ) k ; x = 7 +
9t , y = 7t , z = 10 + 8t
r = ( 11 + 8t ) i + 6t j + ( 14 + 7t ) k ; x =
11 + 8t , y = 6t , z = 14 + 7t
Date: _____________
10 Which of the following pairs of points lies on a line that is
parallel to the line through points ( 2, 1, 2)
and
( 5, 2, 4)
?
a.
b.
c.
d.
e.
( 8, 5, 7), ( 11, 6, 9)
( 12, 9, 11), ( 12, 7, 10)
( 9, 6, 8), ( 9, 4, 7)
( 5, 2, 4), ( 15, 10, 13)
( 6, 3, 5), ( 8, 3, 6)
11 Find parametric equations for the line through ( 4, 2, 0)
that is perpendicular to the plane 5x y + z = 1.
a. x = 4 + 5t , y = 2 t , z = t
b. x = 2 t , y = 4 + 5t , z = t
c. x = t , y = 4 + 5t , z = 2 t
d. x = t + 20, y = 4 + 5t , z = 2 t
e. x = 4 + 5t , y = t , z = 2 t
12 Which of the following lines is parallel to the line with
parametric equations x = 30t , y = 6 + 48t ,
z = 24t ?
a. x = 4 + 5s , y = 1 4s , z = 8s
b. x = 1 + 5s , y = 4 8s , z = 4s
c. x = 6 + 5s , y = 1 4s , z = 8s
d. x = 4 + 8s , y = 1 5s , z = 4s
e. x = 6 + 4s , y = 4 8s , z = 5s
13 Find an equation of the plane through the point
( 2, 0, 2) and with normal vector 5j + 9k .
a.
b.
c.
d.
e.
6y + 10z = 19
4x + 8y + 12z = 21
7y 11z = 20
5y + 9z = 18
2x 5y + 9z = 18
PAGE 2

Name: __________________
14 Find an equation of the plane through the points
( 0, 7, 2) , ( 2, 0, 2) , and ( 5, 7, 0)
.
a.
b.
c.
d.
e.
16x + 4y 35z = 93
16x + 4y + 35z = 106
14x + 4y + 35z = 98
14x 4y + 35z = 97
15x 4y + 35z = 93
15 Find an equation of the plane that passes through the
point ( 2, 6, 7) and contains the line x = 6t , y =
3 + t , z = 5 t .
a.
b.
c.
d.
e.
14x 5y + 16z = 39
5x + 14y + 16z = 36
5x 14y + 16z = 38
16x + 5y + 14z = 41
14x 5y + 16z = 36
16 Find the point at which the line x = 9 + 5t , y = 6t ,
z = 14 7t intersects the plane x + 4y z +
5 = 0.
17 Which of the following planes is parallel to the plane
5x 4y + 9z = 4?
a.
b.
c.
d.
e.
10x 8y + 7z = 3
10x + 8y + 7z = 3
45x 25y 20z = 3
25x + 20y 45z = 3
25x 20y + 45z = 3
18 Find the point at which the lines
r 1
= 7, 8, 0 +
t 1, 1, 9 and r = 15, 0, 54 +
2
s 1, 1, 0
2, 8, 0 + t 1, 1, 6 and r = 2
10, 0, 48 + s 1, 1, 0
a. x + y = 11 d. x + y = 13
b. x + y = 12 e. x + y = 3
x + y = 10
intersect:
19 Find an equation of the plane that contains the lines
c.
.
Class: Math 22
Spring 2005
r 1
=
Date: _____________
20
5z = x 2 Find the traces of the surface y 2
in the
planes x = k , y = k , and z = k .
a.
b.
c.
d.
e.
x = k : 5z + k 2 = y 2 ; y = k : 5z k 2 =
x 2 ; z = k : x 2 y 2 = 5k
x = k : 5z k 2 = y 2 ; y = k : 5z + k 2 = x 2
;
z = k : x 2 + y 2 = 5k
x = k : 5z k 2 = y 2 ; y = k : 5z + k 2 =
x 2 ; z = k : x 2 y 2 = 5k
x = k : 5z k 2 = y 2 ; y = k : 5z + k 2 =
x 2 ; z = k : x 2 y 2 = 5k
x = k : 5z k 2 = y 2 ; y = k : 5z k 2 =
x 2 ; z = k : y 2 x 2 = 5k
21 Find an equation for the surface obtained by rotating the
line x = 4y about the x axis.
a.
b.
c.
d.
x 2
16 + y 2 + z 2 = 1
x 2
16 = y 2 + z 2
x 2 + 16y 2 + 16z 2 = 1
x 2 = y 2 + 16z 2
e. z = 16x 2 y 2
22 Find the rectangular coordinates of the point with
cylindrical coordinates 9, 2 , 4 .
a. ( 0, 4, 9) d. ( 9, 0, 4)
b. ( 9, 9, 4) e. ( 9, 0, 4)
c. ( 0, 9, 4)
23 Find the rectangular coordinates of the point with
cylindrical coordinates ( 3, , 4e ) .
a. ( 0, 3, 4e ) d. ( 3, 0, 4e )
b. ( 0, 6, 4e ) e. ( 3, 0, 4e )
c. ( 0, 6, 4e )
PAGE 3

Name: __________________
Class: Math 22
Spring 2005
24 Find cylindrical coordinates of the point with rectangular
coordinates ( 3, 3, 4)
.
Date: _____________
27 Find spherical coordinates of the point with rectangular
coordinates ( 0, 6 3, 6)
.
a.
3 2, 3 4 , 4
a. 6, 3 , 3 d.
2
13, 3 2 , 3
b.
5, 4 , 4
b. 12, 2 , e.
3
13, 2 , 3
c.
3, 4 , 4
c.
12, 3 , 2
d.
3 2, 4 , 4
e.
5, 3 4 , 4
25 Find the rectangular coordinates of the point with spherical
coordinates ( 3, 0, 0)
.
a. ( 0, 0, 3) d. ( 0, 0, 3)
b. ( 0, 3, 0) e. ( 3, 0, 0)
c. ( 0, 3, 0)
26 Find the rectangular coordinates of the point with spherical
coordinates 18, 5 4 , .
3
a.
b.
c.
d.
e.
9 6
4 , 9 6
4 , 18
9 3
4 , 9 3
4 , 9
9 3
2 , 9 3
2 , 9 2
9 6
2 , 9 6
2 , 9 2
9 6
2 , 9 6
2 , 9
PAGE 4