If I have a rotation matrix $R$, say:
$$R = \begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{bmatrix}$$
when I multiply it by a vector,
$$V' = R.V$$
does the vector necessarily have to be a unit vector for my result to be correct?
Best Answer
Consider some arbitrary vector $\mathbf u$. It has a magnitude of $\|\mathbf u\|$. Clearly then, $\hat{\mathbf u} = \dfrac 1{\|\mathbf u\|} \mathbf u$ is the unit vector in the same direction as $\mathbf u$. Let the rotated unit vector be $R\hat{\mathbf u} = \hat{\mathbf u}'$. Now let's consider $R\mathbf u = \mathbf u'$.
$$R\mathbf u = R\frac {\|\mathbf u\|}{\|\mathbf u\|}\mathbf u = \|\mathbf u\|R\dfrac 1{\|\mathbf u\|} \mathbf u = \|\mathbf u\|R\hat {\mathbf u} = \|\mathbf u\|\hat{\mathbf u}' = \mathbf u'$$
So this (hopefully) makes it clear that $R\mathbf u$ is the vector which is rotated by $R$ by an angle $\theta$ and has the same length as $\mathbf u$.