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In 3-dimensional space, we have an explicit formula for the rotation matrix which will rotate about a vector $\vec{a} = [a_x, a_y, a_z]$. This is given by:

$$ \begin{bmatrix} \cos\theta+a_x^2(1-\cos\theta) & a_xa_y-a_z\sin\theta & a_xa_z(1-\cos\theta)+a_y \sin\theta \\ a_xa_y(1-\cos\theta)+a_z\sin\theta & \cos\theta+a_y^2(1-\cos\theta) & a_ya_z(1-\cos\theta)-a_x\sin\theta\\ a_xa_z(1-\cos\theta)-a_y\sin\theta & a_y a_z(1-\cos\theta)+a_x\sin\theta& \cos\theta+a_z^2(1-\cos\theta) \end{bmatrix} $$

In 4-d space, rotations are defined in a plane (in 3-d space as well, the rotation is happening in the plane orthogonal to the vector, $\vec{a}$).

Is there a corresponding formula that takes two vectors spanning the space that rotation is going to happen in (or the spanning the space which is going to be unaffected by the rotation) and the angle, $\theta$ we are going to rotate by as input and returns a $4 \times 4$ matrix?

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  • $\begingroup$ But in $4$-space you can simultaneously rotate two orthogonal planes by arbitrary amounts. Or are you specifying that you fix the orthogonal $2$-plane? It's just a change of basis formula computation. $\endgroup$ May 22, 2022 at 21:40
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    $\begingroup$ You should have a look here $\endgroup$
    – Jean Marie
    May 22, 2022 at 22:46

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Very good question! You may want to check out Givens rotation matrix $G(i,j,\theta)$, which induces rotation of $\theta$ only in the plane defined by $i$ and $j$ axis of the space, with the orthongonal subspace unchanged.

If you do a basis change $P$ such that the first axes $1,2$ correspond to the vectors spanning the rotation plane you defined. Then this shall be the required rotation matrix $$ PG(1,2,\theta)P^{-1} $$

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  • $\begingroup$ Thanks! Follow up: can every rotation in 4-d space be represented like this? $\endgroup$ May 23, 2022 at 2:22

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