Rotate bunch of vectors around a specific vector

I have a group of vectors in $n$-dimensional space. Now I want to rotate all vectors around a specific vector, say $K$, so that:

• the angle between any vector and $K$ be equal by the angle between its rotation vector and $K$
• the angle between any vector and it's rotation be $\theta$

How can we get the rotation formula for this problem? Certainly it should be stated in matrix form so that can apply it to all vectors.

Thank you.

• I'm not sure I under I understand what you want. But rotating points around the origin or any way is a linear transformation, and a matrix. If it isn't around the origin, it requires translation which is only a linear transformation in next dimension where extra coordinate is added. So $\Bbb R^3$ requires a 4 dimensional vector and $4 \times 4$ matrix. – marshal craft Apr 21 '17 at 15:06
• Your second condition can't, in general, be satisfied. Any of the vectors parallel to $K$ won't be rotated. The angle between vectors not normal to $K$ and their rotated version will be between zero and $\theta$. – T L Davis Apr 24 '17 at 23:23