TITLE: Seven Problems on Planes, Lines and Vectors in 3D
# !b1 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:
- The points of intersection between the sphere and the line
- The area of the triangle containning the pts in (1) and
the north pole of the sphere.

Solution1
!b1 Problem2:
Find three different vectors in 3D, u,v,w such that the cross product
between them is not associative.
Solution2
!b1 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
!b1 Problem4:
Show that the line
!c0 (x-1)/3 = -y/2 = z+1
and the plane x + 2y + z = 1 are parallel.
Solution4
!b1 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
!b1 Problem6:
Find the cross product and the inner product between
the vectors (i+j) and (i-j+k).
Solution6
!b1 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
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