# how to find transformation matrix between two given 3d vectors

i've found something about it but with quaternions, idont know anything about quaternions. So it would be great to find solutions in terms of vectors.

I mean rotation matrix..

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Do you mean to ask, given $\vec{a}, \vec{b} \in \mathbb{R}^3$, find $A$ s.t. $A \vec{a} = \vec{b}$? –  gt6989b Oct 31 '12 at 16:09
@gt6989b, yes i mean exactly this –  Yola Apr 9 '13 at 9:42

I'm assuming you are asking that given $\vec{x} = (x_k)_{k=1}^3, \vec{y} = (y_k)_{k=1}^3 \in \mathbb{R}^3$, how do you find $A = (a_{i,j})_{i,j=1}^3 \in \mathbb{R}^{3 \times 3}$ so that $A \vec{x} = \vec{y}$.
In case $\{x_i\}_{i=1}^3$ are all non-zero, a simple diagonal matrix would do: define $a_{i,i} = y_i/x_i$ and let all off-diagonal elements be $0$.
If exactly one of the elements of $\vec{x}$ are zero, say $x_1 = 0$ then let $a_{1,1} = 0$ and $a_{1,2} = y_1/x_2$, leving the rest at 0.
If two of the elements of $\vec{x}$ are zero, say $x_1 = x_3 = 0$, define $a_{1,1} = a_{3,3} = 0$, $a_{1,2}$ as above, and $a_{3,2} = y_3/x_2$ to get what you need.
Finally, if $\vec{x} = \vec{0}$, it is not possible unless $\vec{y} = 0$ as well.