Prove some properties of the vectors (cos θ, -sin θ) and (sin θ, cos θ)

Consider the vectors

$A = (\cos \theta, - \sin \theta), \qquad B = (\sin \theta, \cos \theta) \qquad \text{in } \mathbb{R}^2.$

Proof. First, we show $A$ and $B$ have unit length,
$\begin{align*} \lVert A \rVert &= \sqrt{ \cos^2 \theta + \sin^2 \theta} = \sqrt{1} = 1 \\ \lVert B \rVert &= \sqrt{\sin^2 \theta + \cos^2 \theta} = \sqrt{1} = 1. \end{align*}$

Then,

$A \cdot B = \cos \theta \sin \theta - \sin \theta \cos theta = 0. \qquad \blacksquare$
If $(x,y) = xA + yB$ then we have
$\begin{align*} (x,y) = xA + yB && \implies && x = x\cos \theta + y \sin \theta \\ &&&& y &= -x \sin \theta + y \cos \theta. \end{align*}$

This holds for all $(x,y)$ if $\cos \theta = 1$ . If $\cos \theta \neq 1$ then we must have $x = y = 0$ .