# Sec. 2 Exam ```MVC
Quiz #2
NAME:
Show all appropriate work and thinking to receive full credit.
Calculator allowed. Explain briefly how you are using your calculator.
#1(5 pt's) Write the parametric equations of the line that is perpendicular to the plane with equation
3x  2 y  1 and goes through the point (1,1,1) .
#2(5 pt’s) Write the equation of the plane containing that contains the point (3,5,0) and the line
 x  5  2t

 y  1 t
 z  4t

#3(5 pt's) Prove that for a , b 
3

 

, a  b  a  b  2a  b
#4(6 pt's) Find the parametric equations of a line that is perpendicular to the line given by
 x  1  3t

 y  1  2t and is in the plane given by 2 x  y  3 .
 z  3  2t

#5(6 pt's) Determine whether there is a plane that contains the two lines
 x  3t

 y  1  t . If so, find the equation of this plane. If not, explain.
 z  2  5t

 x  3  3s

 y  1  2s and
 z  4s

Spherical/Cylindrical
 r   sin 

  
 z   cos 

  2  r2  z2

r

 tan  
z

   
Spherical/Cartesian
 x   sin  cos 

 y   sin  sin 
 z   cos 


2
2
2
2
  x y z

x2  y 2

tan



z

y

tan  

x
#6(5 pts) Convert the equation  sin( )  csc( ) into an equivalent equation in Cartesian
coordinates and then sketch its graph. [If you want to clarify your picture you may describe the
graph in words]
#7(6 pts) Describe the region inside the solid cylinder of radius 4 and height 4 pictured below.
Assume that the cylinder is centered on the positive z-axis and “sits on the xy-plane. [For example, a
description of a solid sphere of radius 1 centered at the origin would be   1 in spherical and
0  r  1, 0    2 ,  1  r 2  z  1  r 2 in cylindrical].
a. Using cylindrical coordinates.
4
4
b.Using spherical coordinates.
Do one of the following – your choice. Indicate clearly which one you want graded.
2
#8(3 pt's) Determine whether the function f :
linear function. Explain clearly and completely.

2
defined f ( x, y )  (3x  2 y  1,3x  y ) is a
Remember: A function f : 2  2 is a linear function if and only if, for all u , v 
1. f (ku )  kf (u ) , and
2. f (u  v )  f (u )  f (v )
2
, and k 
#9(3 pts) Show that the area of a triangle, two of whose sides are determined by the vectors
2
1
a b  (a  b ) 2 .
a and b is given by the formula: A 
2

a
b

```