Time derivative of rotating vector

[Wardub]

Trying to teach myself physics and I've run into a problem I don't quite understand.

"The magnitude of dA/dt can be found by the following geometrical argument. The change of A in the time interval t to Δt is"

ΔA = A(t + Δt) - A(t)

And then somehow it gets to
|ΔA| = 2Asin(Δ$\theta$/2)

I just can't see how those are equal. Especially where theta/2 comes in.
This comes out of Kleppner pages 25 and 26, if that helps. Supposedly there's a sketch, but I can't see one.

 

[phyzguy]

Work it out. Suppose at t = 0, A points along the x-axis, so A = (A,0), If you rotate by an amount theta, the A = (A cos(theta),A sin(theta). So Delta A = (A(1-cos(theta)),A sin(theta)). So:

$$|\Delta A| =A \sqrt{(1-\cos(\theta)^2 + \sin(\theta)^2} =A \sqrt{1-2\cos(\theta) + \cos(\theta)^2 + \sin(\theta)^2} =A \sqrt{2-2\cos(\theta)} = 2A \sin(\frac{\theta}{2})$$

This last is a half-angle trig identity:

$$\sin(\frac{\theta}{2}) = \sqrt{\frac{1-\cos(\theta)}{2}}$$

 

[Wardub]

Thanks, I actually just got it myself by drawing a picture and using the Pythagorean theorem. Essentially equivalent to what you did.