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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 plane 2x  z  7 .
#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 2x  y  z  1
( x  2) ( y  1) ( z  3)


intersect the plane given by
1
2
3
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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#4(5 pts) Find the parametric equations of the line in the plane 4 x  2 y  z  0 that is perpendicular
 x  2t  1

to the line given by  y  3t  1 .
 z  t  9

#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.
2
a. If u  n , then u u = u .
b. If a  x  0 for all vectors x 
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3
, then a  0 .
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(4 pts)   4sec  into Cartesian
  4sec   cos   4  z  4
This is a plane 4 units above the xy-plane.
#7(4 pts) sin  (cos   sin  )  cos  into both Cartesian and Cylindrical
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 x  2t  3
 x  s3


#8(5 pt’s) Find the distance between the lines given by  y  t  1 and  y  3s  2
 z  3t  2
 z  3s  1


Let P0 (3,1, 2) be a point on the first line and P1  3,2, 1 be a point on the second.
n   2, 1,3  (1,3,3)  (12, 3,7) is normal to both lines.
 (6,1,1)  (12, 3,7) 
projn ( P0 P1 )  
 (12, 3,7)
 (12, 3,7)  (12, 3,7) 
 408 102 238 

,
,

 101 101 101 
34 202
101
 4.78

P1
n
P0
Concepts:
#9(2 pts) The top extremity of a ladder of length L rests against a vertical wall, while the bottom is
being pulled away. Find parametric equations for the midpoint P of the ladder, using as parameter
the angle θ between the ladder and the horizontal ground.
P

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#10 (3 pts) Use vectors to prove that any triangle inscribed in a semi-circle is a right triangle.
C
A
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O
B
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