Calculus III Exam #1
Fall 2005
Name_____________________________
Show all your work on this paper. Solutions without correct supporting work will not be accepted.
1. Let u  4,1,5 and v  3,2,2 , find:
(6 points each)
a. u  v
b. the angle between u and v (approximate the nearest tenth of a degree)
c. a vector orthogonal to u
d. a unit vector in the direction of v
e. the area of the parallelogram with adjacent sides u and v
2. Find a set of parametric equations for the line that passes through the point (1,2,3) and perpendicular to the
plane given by 3( x  5)  2( y  1)  ( z  4)  0 . (8 points)
3. An object is pulled 8 feet across a floor using a force of 75 pounds. Find the work done if the direction of
the force is 30 degrees above the horizontal.
(6 points)
 3 3 3 3
 , convert to
, ,
4. Given the rectangular coordinates of a point 

4
4
2


(5 points each)
a. cylindrical coordinates
b. spherical coordinates
5. a. Write the equation of the plane containing the points (2,1,1), (0,4,1) and (-2,1,4).
b. Sketch a graph of the plane in part (a).
(4 points)
(8 points)
6. Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with a
initial velocity of 900 feet per second and at an angle of 45˚ above the horizontal.
(10 points)
7. Find the tangential component of acceleration for the position vector given by r (t )  3t i  t j  t 2 k .
(8 points)
8. a. Find the unit tangent vector for r (t )  2 cos t ,2 sin t , t at the point (2,0,0). (6 points)
b. Find a set of parametric equations for the line tangent to the space curve at the point in part (a).
(4 points)
 
9. Find the length of the space curve given by r (t )  3t ,2 cos t ,2 sin t over the interval 0,  .
 2
(6 points)