Distance between point on sphere

Question

I have a sphere and 2 points on it. I know everything about these points to be able to define them in 3 dimensional space with both cartesian coordinate system and polar coordinate system (x, y and z position of the points; angles and radius).

I need to find the shortest distance between this point on the sphere's surface.

Answer 1

Hint:

If $\mathbf {p_1}$ and $\mathbf {p_2}$ are the two points, note that $|\mathbf {p_1}|=|\mathbf {p_2}|=r$ ( the radius of the sphere).

Now find the angle $\alpha$ betveen the two points using the dot product: $$ \alpha=\arccos (\frac{\mathbf {p_1}\cdot \mathbf {p_2}}{r^2}) $$

and the distance between them is $ d=r\alpha$.