Math 2A  Chapter 9 Test Solutions  Fall ’07 

 

1.      Let .  Find two unit vectors parallel to .
SOLN:   So .

2.      What work is done by the force vector  in moving an object from the origin to (2,3,4)?  Assume the force is measured in Newtons and the distances are measured in meters.
ANS:  The displacement vector is  work is .  The component of the work vector parallel to the displacement vector is  where θ is the angle between the vectors.  The magnitude of this vector times the displacement is the work done  joules.

3.      Find the angle between the vectors  and  where the coordinates
are  , and  .
ANS:   

4.      Find the distance between the plane  and the plane .
ANS:  Choose point O(0,0,0) in the first plane and P(7,0,0) in the second plane and note that  is vector normal to these planes.   The distance between the planes is the length of the projection of  onto the normal:

5.      Find an equation for the plane S containing the x-axis and passing through the point P(1, 2, 3) .
ANS:  A plane containing the x-axis is perpendicular to the y-z plane and passing through the origin.  This is the plane .

6.      Consider the quadric surface  described by the equation  

a.       Write the equation in standard form
ANS:  
has no real valued solution.

b.      Identify the surface as one of the following: a hyperbolic paraboloid, an elliptical cone, an elliptical paraboloid, an ellipsoid, a hyperboloid of one sheet or a hyperboloid of two sheets. 
ANS:  It’s none of these.  There is no surface in the space of real numbers.

c.       What is the intersection of the surface with the plane x = 2?
ANS:  There is no intersection.

7.      Find the magnitude of the torque vector at P if a 20 Newton force is applied to the rigid body shown in the diagram at right. Note that this is a planar diagram.   Recall that  where  is the lever arm vector  is the force vector and θ is the angle between them.
ANS:   Joules

8.      Describe the cross section that the given plane makes with the surface  

a.    z = 9

b.      ANS:   is a hyperbola in the plane z = 9 with vertices at ,   and asymptotes along , z= 9.

c.       x =2y
ANS:  If x =2y then  so that the intersection is the line x =2y in the xy-plane.

9.      The spherical coordinates of a point are  

a.    What are the rectangular coordinates?
ANS:  Generally speaking,  needs to be a non-negative angle less than or equal to π, which the given value is not.  Can we still interpret in a sensible manner? How about having the angle swing backwards?  That is, the forwards direction is chosen by θ and then  swings backwards from there.   This is equivalent to subtracting rotating 180° about the z-axis, which can also be achieved by subtracting (or adding) π to θ  so  and


So the rectangular coordinates are  

b.    What are the cylindrical coordinates?
ANS:   and  and
So the cylindrical coordinates are  

10.  Write the equation  

a.       Using cylindrical coordinates.
ANS:   

b.      Using spherical coordinates.
ANS: