How do I find the nearest point on a sphere?

Question

Say I have a sphere of radius $6$ centered at $(3, 4, 5)$.

What's the nearest point on the surface of the sphere to point $(1, 2, 3)$, which is within the sphere?

I feel that this is a minimization problem involving calculus since we can minimize the distance. Is there a way to do this problem without calculus? I want to program it, and using calculus to solve a minimization problem may not be efficient.

Answer 1

Find the unit vector from (3,4,5) to (1,2,3), multiply it by 6, that is the point you are looking for.

Answer 2

Find a line equation passing through the points $(3,4,5)$ and $(1,2,3)$. This line intersect the sphere at two points.(we already know the equation of sphere $(x-3)^2+(y-4)^2+(z-5)^2=6^2$) Chose the point which is closest to $(1,2,3)$ using distance formula.