Math 251 Week in Review Fall 2014 Week 1 J. Lewis
1. A diameter of a sphere is the line segment from P(-1, 4, 6) to Q(3, 2, 8). Find the center, radius and equation of the sphere.
2. Sketch each region in
3
R . a) {(4, y, z): y, z in R } That is, x=4. b) {( x , y , z ) : y
2  z
2 
25 , x
 
2 } c) {( x , y , z ) : y
2  z
2 
25 } d) {( x , y , z ) : x
2 
4 y
2 
16 , z

3 }
3. Sketch the elliptic cylinder, {( x , y , z ) : 9 x
2 
4 y
2 
18 x

16 y

11 ,

1
 z

4 }
4. Describe and sketch each region in
3
R . a ) D
1
: x
2  y
2 
16 b ) D
2
: y
 z

4 c ) D
1

D
2
In c, is it a circle or an ellipse?
5. What is true about the angle between two vectors
 v and
 w if
 v

 w is a) positive b) negative c) 0
6. For
 v

1 ,

1 , 0 ,
 w

2 , 1 , 3 a) find comp
 v
 w and p roj
 v
 w b) Find a unit vector along 3
 v 2

 w .

7. Find the work done if the force F is applied along the line segment from A(4, -6, 2) to B(3, 0, -1).
 a) F

3
 i

2
 j

b) F is the force of gravity on an object weighing 10kg.

8. Find the torque if the force F

3
 i

2
 j is applied to a lever arm parallel to 2
 i

5
 j

 k .
9. Find vector, parametric and symmetric equations of the line:
 a) through P(3, 4, -6) and and parallel to a

2
 i

7
 j

 k . b) through P(3, 4, -6 ) and Q(2, -3, 5).

c) orthogonal to the lines in parts a and b above and through R(2, 5, -4).

10. Given the points A(-2,-1,0) , B(3,1,2), C(-2,2,4) and vector n

4 ,

5 , 3
 a) find the plane with normal vector n

4 ,

5 , 3 containing A. b) find the plane containing the points A, B, and C. c) Find the distance from P(2, 3, 1) to the plane found in b above.
11. Extra problem : Find the distance from the origin to the line through R(-1, 2, -3) and S(2, 7,6).