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Seven Problems on Planes, Lines and Vectors in 3D


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Problem1:

Consider the unit sphere centered at the origin and the line with direction (2,1-3a,2-3a) passing through the point (0,a,a). When a=sqrt(2)/2 Find:
  1. The points of intersection between the sphere and the line
  2. The area of the triangle containning the pts in (1) and the north pole of the sphere.
Solution1

Problem2:

Find three different vectors in 3D, u,v,w such that the cross product between them is not associative. Solution2

Problem3:

Given a line with possition vector u and velocity v, find the coordinates of all the points at a fix distance R from the line. Deduce from there the equation of the cylinder. Solution3

Problem4:

Show that the line
(x-1)/3 = -y/2 = z+1
and the plane x + 2y + z = 1 are parallel. Solution4

Problem5:

Find the formula for the angle between a line and a plane and use it to find the angle (in degrees) between the plane x+y-z = -1 and the line y = 1 - 3x on the xy-plane. Solution5

Problem6:

Find the cross product and the inner product between the vectors (i+j) and (i-j+k). Solution6

Problem7:

Is the distance between to parallel planes, Ax + By + Cz = D1 and Ax + By + Cz = D2 given by |D1-D2| ? If not, what is the correct formula. Find the distance between the planes, x+y+2z=2 and x+y+2z=4. Solution7


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Tue Feb 8 13:45:04 EST 2000