Sorry if this is a dumb question. I'm learning about 3D computer graphics and how to rotate a vector $\mathbf v$ around an arbitrary unit vector axis $\mathbf n$ by angle $\theta$. What I don't get is why $\mathbf v$ is used in the cross product calculation $\mathbf n×\mathbf v$ instead of the perpendicular component $\mathbf v_⊥$, since we're trying to find a vector that's orthogonal to both $\mathbf v_⊥$ and $n$.