Prove the following vector identity:
Any vector a dotted with its time derivative is equal to the vector's scalar magnitude times the vector's derivative's scalar magnitude.
(a)dot(d(a)/dt)=||a|| x ||da/dt||
The Attempt at a Solution
I tried separating into components, but then it hit me. This identity doesn't make any sense. If a-dot represents the time derivative, then it also represents the velocity tangent to the curve formed by a, which is obviously not always going to be in the same direction. The identity presumes that the angle formed between the time derivative and the vector is zero. How can this be accurate?