Explain and sketch whenever possible. Put final answer in a frame. Each problem carries the same weight.
1) Give the point A = (1, 2, 3) and the straight line (x+1/1) = (y-1/2) = z/2. Determine:
a) The (shortest) distance between the point A and the line.
b) An equation of the plane containing A and the given line.
2) Given two lines L1 and L2
L1 = (x+2/0) = (y-4/-2) = (z+1/4), L2 = x/1 = y/3 = z/4.
Determine:
a) Whether or not the two lines intersect.
b) The point of intersection of L1 above and the plane x - y - z =1.
3) Do the two planes-
x - 2y + z =1, x + y + z = 9.
Intersect? If they intersect find an equation of the line of intersection. What is the distance between the two planes?
4) A parallel piped has the following four adjacent vertices A = (0, 0, 0), B = (1, 1, 1), C = (1, 2, 3) and D = (1, 0, 1). Determine:
a) The coordinate of the point Q that makes ABCQ a parallelogram.
b) The area of the triangle ABC.
c) The volume of the parallelepiped.
d) The length of the height that extends from the point D to the plane that contains the three points A, B, C.