Vector Equations

Question

Consider the planes 5x+2y+2z =1 and 5x+2z= 0. How do you find the vector equation for the line of intersection of the two planes?

I was able to find the unique point on the y-axis which is on both planes to be (0,1/2,0) and that the unit vector with the positive first coordinate that is parallel to both planes to be u= (4/sqrt(116)) i + 0 j - (10/(sqrt(116)) k.

The things that I solved for was correct, just I do not know how to find the vector equation for the line of intersection of the two planes. If you could explain, that would be helpful.

Answer 1

This problem is easier than most of this sort, as each equation includes 5x + 2z. If we subtract one equation from the other, we see that 2y = 1, or y = 1/2. Then, as 5x + 2z = 0, 2z = -5x, or z = -5/2x

Thus, as (0, 0) is the easiest point to know works for (x, z), as z = -5/2x, our solution is simply

(0, 1/2, 0) + (1, 0, -5/2) t or (t, 1/2, -5/2t)