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SHOW ALL WORK !!
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Quiz #2
Name:
#1(4 pts) Find the equation of the plane through (1, 3, 4) that is parallel to the xy-plane.
#2(5 pts) Find the parametric equations of the line of intersection of the two planes given by:
2 x  3 y  z  7 and x  y  2
( x  2) ( y  1) ( z  3)


intersect the plane given by
1
2
7
3x  5 y  z  2 . If not, explain how you know. If so, find the point of intersection.
#3(4 pts) Does the line given by
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 x  2t  1

#4(5 pts) Find the angle between the line given by  y  2t  1 and the plane given by
 z t 3

4 x  2 y  z  0 . Remember that the angle is the smallest angle.
#5(8 pt’s) Determine whether each statement is true or false. If true, prove it. If false, give an
example that demonstrates it is false.
If u and v are vectors in
n
b. If a and b are vectors in
3
a.
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and a 
, then a(u v )  au av .
such that a  x  b  x for every vector x in
3
, then a  b .
Spherical/Cylindrical
Spherical/Cartesian

2
2
2
2
  x y z
 x   sin  cos 
 r   sin 

x2  y 2



y


sin

sin

tan








z
 z   cos 

 z   cos 


y

tan  

x
For each of the following, translate the given equation into the specified coordinate system, and
describe its graph precisely. E.g. A sphere is not precise. A sphere of radius 2 centered at (1,2,3)
is precise. You may use the coordinate axes to help describe the graph if you feel it contributes to
a good description.
  2  r2  z2

r

 tan  
z





#6.   sin  cos  into Cartesian
#7. z  x 2  y 2 into both Cylindrical and Spherical
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 x  2t  3

#8(5 pt’s) Find the distance between the line given by  y  t  1 and the plane 2 x  2 y  2 z  3 .
 z  3t  2

Show all work/explain.
Concepts:
#9(3 pts) Use vectors to prove that the triangle formed by connecting the midpoints of the sides
of an isosceles triangle is itself isosceles.
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#10(3 pts) An epicycloid is the path traced out by a point on a circle of radius r as it rolls around
the outside of a fixed circle of radius R. Place the fixed circle with center at the origin and place
the moving circle initially so that it and the fixed circle are tangent at the point ( R,0) and by
having it roll counter-clockwise in the xy- plane. Parameterize the path of P in terms of the angle
 from the positive x-axis to the center of rolling circle.
P

P
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B