Can you help me prove this vector calculus derivative property?

Question

If r(t) * r(t) = c, then r(t) * r'(t) = 0.

r(t) is a vector valued function, and r'(t) is its derivative. c is a constant. The * indicates dot product (not sure what else to use).

I don't need a whole proof (it would be nice though), but an idea of where to start would be very helpful. A proof in two dimensions (as opposed to three) is all I need. Thanks in advance.

Answer 1

r(t) = (x(t) ; y(t))

r(t) * r(t) = (x(t))² + (y(t))² = c
derivating :

2x(t) x'(t) + 2y(t) y'(t) = 0

or x(t) x'(t) + y(t) y'(t) = 0

or r(t) * r'(t) = 0

Answer 2

r(t) * r(t) = c rx(t)rx(t) + ry(t)ry(t)= c rx(t)^2 + ry(t)^2= c, where rx(t) and ry(t) are scalar functions, the components of vector function r(t) Find derivative of this equation, 2rx(t)rx'(t) + 2ry(t)ry'(t)= 0, because c is a constant. Now realize this is just scalar product, r(t) * r'(t) = 0, where the derivative vector has components r'(t) = ( rx'(t), ry'(t) ).