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I have a 3d cartesian coordinates system and now I want to rotate a point $p(x_0, y_0, z_0)$ arround a specified axis $v(v_x, v_y, v_z)$ like $(1,1,1)$,and the angle is $\theta$,finally I want to get the new location of $p$. Can somebody solve it by using the acknowledge of linear algebra?

Can you give me any hints?

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up vote 1 down vote accepted

Extend $v$ to an equinorm orthogonal basis, i.e. find an arbitrary $w\perp v$ with $|w|=|v|$ and then similarly a $u$, s.t. $|u|=|v|$ and $u\perp v$, $u\perp w$. Then write up the rotation matrix in basis $v,w,u$: $$\begin{bmatrix} 1&0&0\\0&\cos\theta&-\sin\theta\\ 0&\sin\theta&\cos\theta \end{bmatrix}$$ and transform it back to the standard basis, by multiplying it by $B:=[u|v|w]$ from the right and $B^{-1}$ from the left.

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Use rotation matrix:

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The translation can be made easier with homogeneous coordinates. – Daryl Oct 3 '12 at 11:12

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