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I've two vectors $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$. How to find transformation matrix for transform from a to b?

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You need some restrictions. For example if a=(0,0,0) and b is anything else, there's no possible transformation matrix. – coffeemath Oct 9 '12 at 17:22
    
@coffeemath sure, you can check my answer, in that case one will not be able to find axis of rotation (and angle of course). – Yola Feb 18 at 16:37

Try the dyadic product $$ \mathbf{a b} \equiv \mathbf{a}\otimes\mathbf{b} \equiv \mathbf{a b}^\mathrm{T} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix} = \begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3 \end{pmatrix}. $$ Maybe you should include a normalization factor.

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and interchange a and b... – draks ... Oct 9 '12 at 10:28
up vote 0 down vote accepted

One can use axis-angle representation to get the rotation matrix.

$$Rot(u,\phi)=I \cos\phi + u u^T(1-\cos\phi) + \hat{u}\sin\phi$$ where $u$ is unit axis vector and $\phi$ is angle of rotation.

To find axis vector you need to use cross product of given two vectors $(\overline a\times\overline b)$, and from this cross product you can get first $\sin\phi$ and then $\cos\phi$.

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