MATH 200 Study Guide - Quiz Guide: Unit Vector, Parallelogram
MATH 200 104 Quiz 1 Name:
1. (2 points) Find the equation of the sphere centered at (2,3,−3) whose intersection with the
plane x-zplane is a circle with radius 4.
(2,3,−3)
The equation of the sphere is
(x−2)2+ (y−3)2+ (z+ 3)2=r2.
To find the intersection with the x-zplane set y= 0 (the equation of the x-zplane).
(x−2)2+ (0 −3)2+ (z+ 3)2=r2
(x−2)2+ (z+ 3)2=r2−9
This is an equation of a circle with radius2equal to r2−9. To have this radius be 4 solve
r2−9=42
so r= 5.
Answer: (x−2)2+ (y−3)2+ (z+ 3)2= 25
2. (2 points) Find the unit vectors with the indicated direction.
(a) Perpendicular to the x-yplane.
There are two right answers.
Answer: ~
k=<0,0,1>or −~
k=<0,0,−1>
(b) Parallel to the vector ~
i−~
j−~
k.
There are two right answers. Both are unit vectors ±~
i−~
j−~
k
||~
i−~
j−~
k|| where ||~
i−~
j−~
k|| =√3.
Answer: 1
√3(
~
i−~
j−~
k) or −1
√3(
~
i−~
j−~
k)
1
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Document Summary
Name: (2 points) find the equation of the sphere centered at (2, 3, 3) whose intersection with the plane x-z plane is a circle with radius 4. (2, 3, This is an equation of a circle with radius2 equal to r2 9. To have this radius be 4 solve (x 2)2 + (z + 3)2 = r2 9 r2 9 = 42 so r = 5. Answer: (x 2)2 + (y 3)2 + (z + 3)2 = 25: (2 points) find the unit vectors with the indicated direction. (a) perpendicular to the x-y plane. Answer: ~k =< 0, 0, 1 > or ~k =< 0, 0, 1 > (b) parallel to the vector ~i ~j ~k. 1 3 (~i ~j ~k) or 1 3 (~i ~j ~k) ||~i ~j ~k|| where ||~i ~j ~k|| = 3. 1: (4 points) let p = (0, 3, 0), q = (2, 2, 2) and ~v = ~j + ~k.
