General question about a derivation

wolfmanzak

I just have a question of "why/how?" I know that for instance [tex]\mathbf v=\omega \hat k \times \mathbf r[/tex] where [tex]\mathbf v[/tex] is my vector for velocity, [tex]\omega[/tex] is my angular velocity and [tex]\mathbf r[/tex] 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 [tex] v= \omega r[/tex]

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.

[tex] v_{B}= \omega_{B|A} r_{A}[/tex]

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.