I am attempting to derive the following formula for rotation of a vector $\mathbf{u}$, undergoing a left-handed rotation $\mu$, to achieve a new vector $\mathbf{v}$. The expression is as follows:
$\mathbf{v} = (1-\cos{\mu})\langle\mathbf{u},\mathbf{n}\rangle\mathbf{n} + (\cos{\mu})\mathbf{u} - \sin{\mu}(\mathbf{n}\times\mathbf{u})$
Here is a diagram of what is going on:
The textbook writes the following:
$\mathbf{v} = \vec{ON} + \vec{NW} + \vec{WV}$
$\mathbf{v} = \langle\mathbf{u},\mathbf{n}\rangle\mathbf{n} + \frac{\mathbf{u} - \langle\mathbf{u},\mathbf{n}\rangle\mathbf{n}}{|\mathbf{u}-\langle\mathbf{u},\mathbf{n}\rangle\mathbf{n}|}NV\cos{\mu} + \frac{\mathbf{u}\times\mathbf{n}}{|\mathbf{u}|\sin{\phi}}NV\sin{\mu}$
Note that $\phi$ was never defined in the text, which makes this slightly frustrating.
I am trying to understand these three terms; they will simplify to the formula I want. I get that the first term is the projection of $\vec{ON}$ onto $\mathbf{u}$, but I am not sure of what the other two terms mean. Any help would be much appreciated.
(for more context, see page 8 of the pdf file here - this gives the derivation, but no explanation of what each term is. This page is an excerpt from my textbook.)