How do you rotate a vector on a plane formed by two vectors other than the first vector in n-dimensions. I know that you can extract the equation of a plane in n-dimension using the two vectors, since the plane will be a linear combination of those two vectors. But I am wondering how would I get a vector other than those involved in plane formation to rotate on that plane with a given angle of rotation.
I am stuck on this thought, and have tried few things, but they are not satisfactory. Any help/insight is much appreciated.
 A: One way to handle this easily is writing the rotation using an orthonormal basis of your plane. That is, suppose your plane is generated by the linearly independent vectors $\mathbf{v}$ and $\mathbf{w}$ of the Euclidean space $\mathbb{R}^n$, then first transform this basis to an orthonormal one, by example setting
$$
\mathrm{\mathbf e}_{1}:=\frac{\mathbf{v}}{\|\mathbf{v}\|},\quad \mathrm{\mathbf e}_{2}:=\frac{\mathbf{w}-\langle \mathbf{w},\mathrm{\mathbf e}_{1} \rangle \mathrm{\mathbf e}_{1} }{\|\mathbf{w}-\langle \mathbf{w},\mathrm{\mathbf e}_{1} \rangle \mathrm{\mathbf e}_{1}\|}\tag1
$$
where $\langle \,\cdot\, ,\,\cdot\,  \rangle$ is the dot product and $\|\mathbf{r}\|:=\sqrt{\langle \mathbf{r},\mathbf{r} \rangle }$ is the induced norm. Then any vector on the plane generated by $\mathbf{v}$ and $\mathbf{w}$ can be written as $\lambda \mathrm{\mathbf e}_{1}+\mu \mathrm{\mathbf e}_{2}$ for scalars $\lambda ,\mu\in \mathbb{R}$, so using the ordered basis $B:=\{\mathrm{\mathbf e}_{1},\mathrm{\mathbf e}_{2}\}$ you can write these vectors as $(\lambda ,\mu)_B$.
Now to rotate this vector you only need to rotate it in the same way as you will do in $\mathbb{R}^2$, by example to rotate an angle $\alpha $ counterclockwise (respect to the orientation given by $B$) you do
$$
\begin{bmatrix}
\cos \alpha & -\sin \alpha \\\sin \alpha &\cos \alpha 
\end{bmatrix}\begin{bmatrix}
\lambda \\\mu
\end{bmatrix}\tag2
$$
Now the result of (2) will be the rotated vector written in basis $B$.
