Two points on the sphere S of radius 1 have spherical coordinates P: \phi = 0.35 \pi, \theta = 0.8 \pi and Q: \phi = 0.7 \pi, \theta = 0.75 \pi. Find a vector parallel to the line of intersection of the tangent planes to S at the points P and Q.
2. The attempt at a solution
First, I converted spherical to cartesian coordinates.
P (-.72083942, .26684892, -.80901699)
Q (-.5720614, -.41562694, -.70710678)
Then, I found two tangent planes at P and Q. To do this, I found the gradient of the general sphere equation x^2 + y^2 +z^2 = 1. And then the general equation of the tangent plane for each point.
Tangent P = -1.44167x + .533698y - 1.161803z -2.4906521
Tangent Q = -1.1441228x - .83125388y - 1.4142136z -2
3. Relevant equations
I'm lost on what to do after that though. I talked to a tutor about this question and he said to set the two equation so that they equal z. Then add them together to get a single equation of a line. And then use parametrics since it's asking for a vector parallel. Does this sound right?