User:Corkgkagj/sb

Let $$V_{\alpha,\,\beta}^2$$ signify the normed vector space whose basic coordinate vectors are $$\boldsymbol{\alpha}$$ and $$\boldsymbol{\beta}$$; and let $$V_{i,\,j}^2$$ signify the normed vector space whose basic coordinate vectors are $$\mathbf{i}$$ and $$\mathbf{j}$$ such that $$\begin{align} \boldsymbol{\alpha}&=\cos{\theta}\mathbf{i}+\sin{\theta}\mathbf{j}\\ \boldsymbol{\beta}&=-\sin{\theta}\mathbf{i}+\cos{\theta}\mathbf{j} \end{align} \qquad \begin{align} \mathbf{i}&=\cos{\theta}\boldsymbol{\alpha}-\sin{\theta}\boldsymbol{\beta}\\ \mathbf{j}&=\sin{\theta}\boldsymbol{\alpha}+ \cos{\theta}\boldsymbol{\beta} \end{align}$$