Invariance of the Dot Product Under Rotation
Let \(R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\) be a rotation matrix in \(\mathbb{R}^2\). Given two arbitrary vectors \(\mathbf{v}\) and \(\mathbf{w}\) in \(\mathbb{R}^2\), prove that the dot product is invariant under rotation by showing that \((R(\theta)\mathbf{v}) \cdot (R(\theta)\mathbf{w}) = \mathbf{v} \cdot \mathbf{w}\).