So I have a 3D vector math problem that I'm having difficulty solving. Basically I have two known vectors in the form (x,y,z), let's call them C and P, and I want to find a third unknown vector, let's call it A.
I know the angle between C and A is 60 and the angle between P and A is 90. I also know that C and P are unit vectors already and that my unknown A is going to be a unit vector. I also know that I already have two solutions to the problem (as opposed to 0 or 1 possible solutions depending). I formulated it as follows and I think this is correct. So I have:
C=(0, 0, 1)
P=(0.61, 0.61, 0.5)
A=(x, y, z)
This gives me the following three equations which should allow me to find both solutions to my three variables but I'm not getting results that make any sense.
$ \cos(90) = 0.61x + 0.61y + 0.5z $ $ \cos(60) = 0x + 0y + 1z $ $ \sqrt{x^2 + y^2 + z^2 } = 1 $
Finding z is easy as it must be 0.5 (which makes sense as z must be fixed for both solution in my problem). I've tried solving using substitution from here to find a quadratic equation for y, finding both y values, then using substitution again to find corresponding x values. I'm not getting any useful results however as the results aren't unit vectors and are therefore incorrect.
For clarity sake I'm trying to find the intersection points between a great circle and small circle on a 3D sphere. P represents the normal vector to the plane that forms the great circle and N represents the centre of the 60 degree small circle on the sphere.