Perpendicular Vector Derivatives and Lengths

I'm learning vectors. I read somewhere that if a vectors magnitude is constant, then its derivative is perpendicular. However, in polar co-ordinates, I learnt something else. The distance of a particle in orbit from a focus is r. if /r/ varies with t even then dr/dt is v and it is perpendicular to r. Further if /v/ varies, then dv/dt is a and is is perpendicular to v and parallel to r. Must the magnitude of the vector be constant or not for the derivative to be perpendicular or not?
- Arman (age 18)
Vancouver, BC, Canada

Yes, the length is constant if and only if the derivative is perpendicular to the vector. You can see that by drawing a picture of a vector and adding to it another tiny vector. If the added piece has anything parallel to the main piece, then it will either make it longer or shorter. The choice of coordinate descriptions has nothing to do with it.

Mike W.

(published on 12/29/2013)