Practice Problems
Math 2500
Feb. 1, 2008
These problems are not in any particular order. The exam will be shorter (about or 5 problems). Keep in mind that
these problems are meant to be representative, and not an exhaustive list of problems for the exam.
1. Given the following pairs of vectors ~u and ~v, find the angle θbetween them and compute the cross product
~u ×~v.
(a) ~u = (1,2,3), ~v = (−2,1,0)
(b) ~u = (1,0,2), ~v = (2,1,0)
(c) ~u = (1,1,0), ~v = (1,0,1)
2. Consider the plane Π1containing p= (2,1,3), with normal vector ~n = (−1,2,0).
(a) Write down the linear equation any point (x, y, z) in this plane must satisfy.
(b) Find the angle between the plane Π1and the plane Π2determined by x−y= 2.
(c) Parameterize the line lwhich is the intersection of Π1and Π2.
(d) Find the distance between the plane Π1and the point q= (3,3,3).
3. Consider the vectors ~u = (1,2,1) and ~v = (0,1,−1).
(a) Explain why all the planes parallel to both ~u and ~v will have the same normal vectors (up to scaling).
(b) Are any of these planes parallel to the plane given by x+y+z= 2? Explain your answer.
4. Consider the plane curve given by c(t) = (cos(t),sin(2t)), for 0 ≤t≤2π.
(a) Sketch this curve.
(b) Set up, but do not evaluate, the integral to compute the arclength of c.
(c) Notice cis periodic (c(0) = c(2π)). Is ca simple closed curve? In othre words, are the tparameters 0 and
2πthe only times ccrosses itself?
5. (a) Consider the right circular cone C, with vertex at (0,0,0), and slope 1. In other words, the cone Cis
what you get when you rotate the line y=zin the y−zplane about the z-axis. Write Cin cylindrical
coordinates.
(b) Write the part of the shell 1 ≤x2+y2+z2≤4 lying in the x < 0, y > 0, z < 0 octant in spherical
coordinates.
6. Consider the space curve c(t) = (cos(t),sin(t), t).
(a) Is the velocity vector ever tangent to the x-axis?
(b) Verify the Fundamental Theorem of Calculus by checking
c(2π)−c(0) = Z2π
0
c0(t)dt.
(c) Compute the arclength of cfor 0 ≤t≤2π.
7. Consider the vectors ~a = (2,1,3) and ~
b= (−1,0,1).
(a) Compute the cross product ~a ×~
b.
(b) Find a vector ~x which is perpendicular to ~a and verify that ~x ⊥~a. (There are many correct answers.)
(c) Write ~a as a sum ~a =~u +~v where ~u is parallel to ~
band ~v is perpendicular to ~
b. (Hint: you only need to
find one of ~u and ~v. It might help to draw a picture.)
8. Consider the curve c(t) given by
c(t) = (tcos t, t sin t, t).
(a) Find the velocity and acceleration vectors of this curve.
(b) Is the tangent line to cever parallel to the x−yplane? Be sure to explain your answer.
