MVC
SHOW ALL WORK !!
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Quiz #2
Name: Key
#1(4 pts) Find the equation of the plane through (1, 3, 4) that is parallel to the xy-plane.
The xy-plane has equation z  0 . Any plane parallel to this plane has equation z  constant .
In this case:
z  4
#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
2, 3,1  1, 1,0  1,1,1
The cross product of the normal vectors 
is parallel to the line of
(0,
2,13)
intersection. We note that the point
is on both planes, and hence the line. So the equations
are:
 xt

yt2
 z  t  13

( x  2) ( y  1) ( z  3)


1
2
7
#3(4 pts) Does the line given by
intersect the plane given by
3x  5 y  z  2 . If not, explain how you know. If so, find the point of intersection.
1,2,7  3, 5,1  0
The line is parallel to the plane since the dot product 
. We need only check that
 2, 1,3 is on the line, but not the plane, the line does
the line is not in the plane. Since the point
not intersect the plane.
MVC
 x  2t  1

 y  2t  1
 z t 3

#4(5 pts) Find the angle between the line given by
4 x  2 y  z  0 . Remember that the angle is the smallest angle.
and the plane given by
The angle between the normal vector to the plane and the vector parallel to the line is
complementary to the angle between the line and the plane.
cos 
(4,2, 1) (2, 2,1)
21  9
   77.4
n
 
   12.6 or .21 radians
#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.
a.
If u and v are vectors in
n
and a 
, then a(u v )  au av .
False: Let u  (1,1) , v  (1, 2),and a  2 . Then a(u v )  2, but au av  4 .
b. If a and b are vectors in
3
such that a  x  b  x for every vector x in
3
, then a  b .
True:
ˆ
Proof: Let a  (a1,a2 ,a3 ) and b  (b1,b2 ,b3 ) . If x  i , then a  x  b  x  a1  b1.
ˆ
ˆ
Similarly, x  j  a2  b2 and x  k  a3  b3 .
MVC
Spherical/Cylindrical
Spherical/Cartesian

2
2
2
2
  x y z
 x   sin  cos 
  2  r2  z2

x2  y 2
r


sin





y


sin

sin

tan




r


z
  
 tan  
 z   cos 

z

y
 z   cos 


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.
#6.   sin  cos  into Cartesian
  sin  cos    2   sin  cos 
 x2  y 2  z 2  x
2
1
1

  x    y2  z2 
2
4

1

 ,0,0 

This is a sphere of radius ½ centered at  2
2
2
#7. z  x  y into both Cylindrical and Spherical
Cylindrical:
z  x2  y 2  z  r 2
Spherical:
z  x2  y 2  z  z 2  x2  y 2  z 2
  cos    2 cos 2    2
 cos    cos 2    or  
MVC
cos 
1  cos 2 
 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.
Since (2, 2, 2) (2, 1,3)  4  0 , the line is not parallel to the plane. Hence the line intersects the
plane and the distance between them is 0.
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.
MVC
#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