# Math 251 Week in Review Fall 2014 ... J. Lewis

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

R

3 . 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

R

3 .

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).