Math 251 Chapter 11 Review
J. Lewis
1. Given the 3 points O(0,0,0), P(2,5,3) and Q(4,8,7):
a) Find the area of the parallelogram with OP and OQ as 2 adjacent edges.
b) Find two unit vectors that are orthogonal to the plane containing O, P and Q.
2. a) Find the equation of the plane containing the point P(4,0,1) and the line with symmetric
equations
x 1
2
y2

3
z .

b) Find the distance from the line with vector equation r ( t )  0,6,1  t 2,3,1 to the plane found
in part a.
3. Each quadric surface below is the result of rotating a curve about a coordinate axis.
Fill in the table.
Surface
Curve
axis of rotation
name of surface/graph
x 2  y 2  z 2  1
 x2  y2  z2  1
x2  y2  z  0
Write an equation for the quadric surface we learned which is not a surface of revolution.
4. f ( x , y , z ) 
x2
2
 y2 
2
9
z 2 For each value of c, name the level surface f(x,y,z)=c
and describe the cross sections in the planes y=2 and z=2.
i) c = -1 ii) c=0
iii) c=1
5. Given P(2,5,3) , Q(1,-6,9) and R(6,1,2):
a) Find the point on PQ that is 1/3 of the way from P to Q.

b) Find comp

PR
PQ
c) Find the point on the line through P and Q that is closest to R.
6. Find parametric equations for the tangent line to the curve with vector equation:

a)
r ( t )  e 2 t , te 2 t , t 2 e 2 t at the point P(1,0,0).

b)
r ( t )  t 3 , t 4  1, t 2
at the point P(-8,17,4).
7. Find the unit tangent and normal vectors and verify they are orthogonal.

r (t )  e 2 t cos t , e 2 t sin t , e 2 t
8. a) Find the length of the curve in 7 for 0  t  2 .

b) Find the length of the curve, r (t ) 
2 52 2 32 2 2
t , t ,
t
5
3
2
0  t  1.
9. Sketch the region bounded by the given surfaces.
a) x  y  9 and z  0 and x  z  4 Label the points where x  3, x  3, x  0 .
2
2
b) Inside both x  y  z  50 and z 
intersection.
2
2
x 2  y 2 . Give the center and radius of the
2
c) Inside both 2 z  1  x  y and x  y  z  25 . Give the center and radius of the
intersection.
2
2
2
d) Inside both x  y  z  6 x and x 
intersection.
2
2
2
2
2
y 2  z 2 . Give the center and radius of the