Suppose $\mathbb{v}_1\in\mathbb{R}^{2n}$ and $\mathbb{v}_2\in\mathbb{R}^{2n}$ are two vectors. Consider a hyperplane $P$ in $\mathbb{R}^{2n}$ space that is perpendicular to $(\mathbf{v_1}-\mathbf{v_2})$, so that for any vector $\mathbf{z}$ in $P$ we have $\mathbf{z}.(\mathbf{v_1}-\mathbf{v_2})=0$. Now assume $C$ is the intersection of the hyperplane $P$ with $(\mathbf{v_1}-\mathbf{v_2})$. So the vector $\mathbf{w}=OC$ is also perpendicular to $(\mathbf{v_1}-\mathbf{v_2})$. I want to rotate the vector $\mathbf{w}$ on the hyperplane $P$ with a rotation angle $\alpha$ to find vector $\boldsymbol{\theta}$ (so $\boldsymbol{\theta}.(\mathbf{v_1}-\mathbf{v_2})=0$). How can I find the vector $\boldsymbol{\theta}$? Can I do the rotation with matrix operation?
Additional comment; $P$ is perpendicular to $\mathbf{v}_1-\mathbf{v}_2$ and passes through the origin.
Comment: Let me be more clear about this problem:
I found $\mathbf{w}=[w_1,w_2,\dots,w_{2n}]$ as: $$w_{i}=\frac{1}{\left\Vert \mathbf{v}_{1}-\mathbf{v}_{2}\right\Vert _{2}^{2}}\left(v_{1}^{(i)}\left(-\mathbf{v}_{2}.(\mathbf{v}_{1}-\mathbf{v}_{2})\right)+v_{2}^{(i)}\left(\mathbf{v}_{1}.(\mathbf{v}_{1}-\mathbf{v}_{2})\right)\right).$$ But I need to find the vector $\boldsymbol{\theta}$ which is orthogonal to $\mathbf{v}_1-\mathbf{v}_2$ and has the property that $$\theta_i^2+\theta_{n+i}^2=1,\; \mathrm{for}\; i=1,\dots,n.$$ So, I want to know if there is any relation between $\boldsymbol\theta$ and $\mathbf{w}$? Can I find $\boldsymbol\theta$ from $\mathbf{w}$?