Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

  Printable Key

PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true explain why.

A.     (15 Points)  If A, B, and C are points, v is the vector from A to B, w is the vector from B to C, and v x w  =  0 , then A, B and C are collinear.

B.      (15 Points) If x  =  x(t),  y  =  y(t) are parametric equations of a line then dx/dt  is a constant.  


PROBLEM 2 (21 Points)

Consider the surface x2 + z2 - e2y  =  0 .  This surface is formed by revolving a generating curve about an axis.  Find an equation of this generating curve and state the axis of revolution.


PROBLEM 3  (21 Points) 

Use vectors to find the equation of the line that passes through the point (2,3,4) and is perpendicular to the plane 5x - 4y + 2z = 7.



PROBLEM 4  (21 Points) 

Find all points (if any) of horizontal and vertical tangency.  Make sure to present your answer by listing the points not just the values of q.

        x  =  cos q        y  =  2sin(2q)


PROBLEM 5 (21 Points) 

Determine the area of the first quadrant loop of  r  =  3sin(2q)


PROBLEM 6  (21 Points) 

Show that the polar equation for the hyperbola

           x2          y2
                -            =  1          
           a2          b2  


         r2  =                      
                      1 - e2 cos2 q 

given that 

         e2  =  1 +            


PROBLEM 7  (21 Points) 

Use vectors to determine if the triangle with vertices (1,0,1), (2,1,0), (0,0,4) is a right triangle.



PROBLEM 8  (21 Points) 

Find parametric equations for the a particle moves along the line through (1,4,2) and (3,5,7) such that it is at the point (1,4,2) when t = 0 is at the point (3,5,7) when t = 2 and is speeding up as time progresses