How to show the formula for the sine of two vectors with linear complex structure

Question

How one can show easily the following formula? $\displaystyle \sin{\theta} = \frac{u\cdot J(v)}{||u||||J(v)||}$ where $J$ is the linear complex structure $J:\mathbb{R^n}\rightarrow \mathbb{R^n}$ such that $J^2 = -I_{\mathbb{R}^n}$ where $I_{\mathbb{R}^n}$ is the identity map on $\mathbb{R}^n$. How does this make sense in $\mathbb{R}^2$ and $\mathbb{R}^3$?

Answer 1

In two dimensions we have $\langle u,Jv\rangle=\det(u,v)$. Now use that $$\det(u,v)^2=\det\begin{pmatrix}\langle u,u\rangle& \langle u,v\rangle\\ \langle v,u\rangle &\langle v,v\rangle\end{pmatrix}$$ together with $\langle u,v\rangle=\|u\|\|v\|\cos(\theta)$ and the Pythagorean theorem.