Calculus III Exam 3b Spring 2000 4/24/2000


Each problem is worth 10 points. Be sure to show all work and justifications for full credit. Please circle all answers and keep your work as legible as possible. No real animals were harmed in the making of this exam.
 
1. Write an equation for a plane orthogonal to the line r = <-2, 3, 1> + t<5, -1, 4>.
 
 
 
2. Find the distance between (3,-5,2) and (7, 1, -2).
 
 
 
3. If a gnat is flying along a course given by r(t) = <cos t, sin t, sin 5t>, find the gnat's velocity and acceleration vectors at time t.
 
 
 
4. Find the unit tangent vector to the curve r(t) = <t, t2, t3> at time t=-1.
 
 
 
5. Find the angle between a diagonal of a cube and a diagonal of one of its faces.
 
 
 
6. Find the equation of the plane with x-intercept a, y-intercept b, and z-intercept c.
 
 
 
7. Prove that for every pair of vectors a and b, a is orthogonal to a × b.
 
 
 
8. Reduce the equation 9x2 + y2 4z2 -18x +2y +16z = 10 to one of the standard forms, classify the surface, and sketch its graph (be sure to label which axis is which!). Include on your graph the coordinates of at least six points on the surface.
 
 
 
9. Ken says "Dude, on this Calc test there was this problem about, like, lines and stuff? And it said, like, to find a line that's parallel to this one plane. So I figured that was totally bogus, 'cause there'd be lots of lines that do that. But I read it again and I guess it gave some other stuff too, but not like two points on the line or a vector on the line, 'cause that I woulda known how to do easy. So doesn't that suck, that they, like, gave me stuff I didn't want to know instead of just asking it the normal way?"
 
Explain (clearly, in terms Ken can follow) what other piece(s) of information might have been given in order to specify exactly one line, and briefly tell Ken how he could find the equation given that information.
 
 
 
10. We mentioned in class that the curvature of every line ought to be zero. Prove it. [Hint: Warm up by writing the vector equation for a particular line of your choice (like the one in problem #1, for instance) and crank through the curvature formula. Once you've warmed up, figure out how to do it for any other line.]
 
 
 
Extra Credit (5 points possible):
 
What does the collection of points whose distances from (3, 0, 0), (0, 0, 0), and (-3, 0, 0) sums to 14 look like?