Math 233 Practice Problems for Exam 1 Fall 2006
1. Consider the line Lthrough points A= (2,1,−1) and B= (5,3,−2). Find the intersec-
tion of the line Land the plane given by 2x−3y+ 4z= 13.
2. Two masses travel through space along space curve described by the two vector functions
r1(t) =< t, 1−t, 3 + t2>, r2(s) =<3−s, s −2, s2>
where tand sare two independent real parameters.
(a) Show that the two space curves intersect by finding the point of intersection and the
parameter values where this occurs.
(b) Find parametric equations for the tangent line to each of the two space curves at the
intersection point.
3. Consider the parallelogram with vertices A, B, C, D such that Band Care adjacent to
A. If A= (2,5,1), B= (3,1,4), D= (5,2,−3), find the point C.
4. 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 area of triangle ABC.
(c) Find the distance dfrom the point Cto the line Lthat contains points Aand B.
5. Find paramteric equations for the line of intersection of the planes x−2y+z= 1 and
2x+y+z= 1.
6. 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?
7. (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.
