General question about a derivation

[wolfmanzak]

I just have a question of "why/how?" I know that for instance $$\mathbf v=\omega \hat k \times \mathbf r$$ where $$\mathbf v$$ is my vector for velocity, $$\omega$$ is my angular velocity and $$\mathbf r$$ is my position vector from a point on the axis of rotation of a wheel to a point on the outer edge of the wheel. I also know that $$v= \omega r$$

But I'd like to understand how it's possible to derive/justify the following from what I have above or if there is another means by which this justification is made. I'm just trying to understand a formula.

$$v_{B}= \omega_{B|A} r_{A}$$

This question came up because I saw the final formula at the bottom used in part to solve for the angular velocity of a wheel rotating about a fixed axis where point "A" was at the center of the wheel and point "B" was along the wheel's edge. I guess I'm just trying to figure out why this equation was used, as I don't see any derivation or reasoning for it in text that I'm using and I wouldn't necessarily have thought to use it like shown if I were solving a similar problem. Any explanation as to why/how or what prompted the book to use the equation in this way would really help my understanding of the topic. Thanks in advance.

 

[Simon Bridge]

Change your vector equation to one that deals only in magnitudes and you'll get the one at the bottom.
The book is not making a definition and leaves some details implicit in order to make the math easier.