Transformation matrix to go from one vector to another

Question

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?

Answer 1

Try using the dyadic product, the definition is $$ \mathbf{a b} \equiv \mathbf{a}\otimes\mathbf{b} \equiv \mathbf{a b}^\mathrm{T} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\begin{pmatrix} b_1 & b_2 & b_3 \end{pmatrix} = \begin{pmatrix} a_1b_1 & a_1b_2 & a_1b_3 \\ a_2b_1 & a_2b_2 & a_2b_3 \\ a_3b_1 & a_3b_2 & a_3b_3 \end{pmatrix}. $$

You can construct the rotation matrix $\mathbf{R}$ you're looking for as following $$ \mathbf{b} = \mathbf{R} \mathbf{a} \\ \\ \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = \frac{1}{3} \begin{pmatrix} \frac{1}{a_1}b_1 & \frac{1}{a_2}b_1 & \frac{1}{a_3}b_1 \\ \frac{1}{a_1}b_2 & \frac{1}{a_2}b_2 & \frac{1}{a_3}b_2 \\ \frac{1}{a_1}b_3 & \frac{1}{a_2}b_3 & \frac{1}{a_3}b_3 \end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \\ \mathbf{R} = \frac{1}{3} \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \begin{pmatrix} \frac{1}{a_1} & \frac{1}{a_2} & \frac{1}{a_3} \end{pmatrix} $$