# Finding a unit vector that is parallel to 2 planes?

http://img200.imageshack.us/img200/1579/screenshot…

Given two planes:
3x + 1y + 2z = 1
3x + 2z = 0
A) Find the unique point P on the Y axis where the planes intersect
B) Find a unit vector u with positive first coordinate that is parallel to both planes: ___i + ___j + ___k
C) Use parts A and B to find a vector equation for the line of intersection of both planes: r(t) = ___i + ___j + ___k
Part A is the easy part. To see where each plane intersects the Y axis, substitute 0 for X and 0 for Z.
This gives you y = 1 for one of the planes, and 0 = 0 for the other plane. The y = 1 tells you that the first plane intersects the Y axis at (0,1,0). The 0 = 0 tells you that the second plane covers the entire Y axis, so it also passes through (0,1,0).
For part B, we take the coefficients and use them to set up a cross product:
i j k
3 1 2
3 0 2
Taking the determinant, we get i * (1*2-0*2) + j * (3*2-3*2) + k * (3*0-3*1) = 2i + 0j -3k. This vector has a length of √((2*2)+(-3*-3)) = √(13). So we have to divide by that to normalize it to a unit vector.
2 √(13) / 13 i + 0 j + 3 √(13) / 13 k
Putting the two together for the vector equation of the line, we get:
r(t) =
(0 + (2 √13/13)t) i
+
(1 + 0t)j
+
(0 + (3 √13/13)t) k
You can leave off the 0+ and the +0t if you want simpler expressions in the boxes.