Math 233 Practice Exam 1 Spring 2006
1. (a) Consider the line Lthrough points A= (2,1,−1) and B= (5,3,−2). Find the
intersection of the line Land the plane given by 2x−3y+ 4z= 13.
(b) Find the distance of the point (2,1,−1) and the plane given by 2x−3y+ 4z= 13.
(c) Consider the parallelogram with vertices A, B, C, D such that Band Care adjacent to
A. If A= (3,5,1), B= (5,1,4), D= (−5,2,−3), find the point C.
2. Consider the points A= (2,1,0), B= (1,0,2) and C= (0,2,1).
(a) Find the orthogonal projection proj −→
AB(
−→
AC) of the vector
−→
AC onto the vector
−→
AB.
(b) Find the point Psuch that
−→
AP =proj−→
AB(
−→
AC).
(c) Find the distance dfrom the point Cto the line Lthat contains points Aand B.
3. (a) Find paramteric equations for the line of intersection of the planes 3x+ 2y−z= 4
and 2x+z= 1.
(b) Let L1denote the line through the points (1,0,1) and (−1,4,1) and let L2denote the
line through the points (2,3,−1) and (4,4,−3). Do the lines L1and L2intersect? If not,
are they skew or parallel?
4. (a) Find the volume of the parallelepiped such that the following four points A= (1,4,2),
B= (3,1,−2), C= (4,3,−3), D= (1,0,−1) are vertices and the vertices B, C, D are all
adjacent to the vertex A.
(b) Find an equation of the plane through A, B, D.
(c) Find the angle between the plane through A, B, C and the xy plane.
5. The velocity vector of a particle moving in space equals v(t) = 2ti+ 2t1/2j+kat any
time t≥0. (a) At the time t= 0 this particle is at the point (−1,5,4). Find the position
vector r(t) of the particle at the time t= 4.
(b) Find an equation of the tangent line to the curve at the time t= 4.
(c) Does the particle ever pass through the point P= (80,41,13) ?
(d) Find the length of the arc traveled from time t= 1 to time t= 2.
6. Consider the vector valued function f(x, y) = 6x3y/(2x4+y4).
(a) Does the limit lim
(x,y)→(0,0)f(x, y) exist? Why or why not?
(b) Compute the second partial derivatives of f(x, y) and verify that fxy =fyx.