Proving Velocity Vector Orthogonal to Position Vector on Sphere Surface

[jessawells]

can someone help me with this problem:

"show that a point is traveling along the surface of a sphere if and only if its velocity vector is orthogonal to its position vector."

I know that it is true intuitively - since in centripetal motion, the velocity is always directed toward the center. but how do you prove this using vector calculus?

 

[HallsofIvy]

can someone help me with this problem:

"show that a point is traveling along the surface of a sphere if and only if its velocity vector is orthogonal to its position vector."

I know that it is true intuitively - since in centripetal motion, the velocity is always directed toward the center. but how do you prove this using vector calculus?

Then your "intuition" is way off. A point moving along the surface of a sphere is not in "centripetal motion" and the velocity vector is not directed toward the center!

You can assume that your sphere is centered at the origin and, so, has equation x²+ y²+ z²= R². Differentiate that with respect to time (using the chain rule) and break the result into a dot product of velocity vector with position vector.

 

[Berislav]

Couldn't the OP just apply curl to the position vector (field) of the sphere? It's seems more in the spirit of vector calculus.