Question

How would you write an equation to represent rotations?

Answer 1

Please nnamely: you can write a rotatio, using matrices:
\[{\mathbf{x'}} = A{\mathbf{x}}\]
where x' is the column of transformed coordinates, x is the column of original coordinates, and A is the rotation matrix, namely:
\[A = \left( {\begin{array}{*{20}{c}}
{\cos \alpha }&{ - \sin \alpha }&0 \\
{\sin \alpha }&{\cos \alpha }&0 \\
0&0&1
\end{array}} \right)\]
which is a rotation around the z-axis, of an angle \[\alpha \]

Answer 2

of, course x' and x are three components vectors