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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.

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  • $\begingroup$ 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. $\endgroup$ – marshal craft Apr 21 '17 at 15:06
  • $\begingroup$ Please read this article everything you need is here, I've used this article to program a renderer in java using matrix multiplication. en.m.wikipedia.org/wiki/Rotation_matrix $\endgroup$ – marshal craft Apr 21 '17 at 15:14
  • $\begingroup$ 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 $\theta$. $\endgroup$ – T L Davis Apr 24 '17 at 23:23

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