Rotating two vectors to point in the same direction

Question

I have two vectors, $v$ and $u$. How do I rotate $u$ around the x-, y-, and z-axes (or one axis) so that it points in the same direction as $v$?

Answer 1

If $u,v\in\mathbb R^2$, Find the angle between $u$ and $v$ by $$\cos\theta=\frac{<u,v>}{|u|.|v|}$$ Now take matrix of rotation $A_\theta$ of angle $\theta$. Now take $A_\theta u$ or $A_{-\theta} u$. These will rotate $u$ to the direction of $v$. $$A_\theta= \left( \begin{array}{cc} \cos\theta &-\sin\theta \\ \sin\theta &\cos\theta \\ \end{array} \right) $$

For $u,v\in \mathbb R^3$,
Write $v= (a,b,c)$ and $u= (x,y,z)$. If all x,y,z is non zero, then we want $T$ such that $Tu = v$ , define $T$ such that $$T(e_1)= \frac{a}{x} e_1$$ $$T(e_2)=\frac{b}{y} e_2$$ $$T(e_3)= \frac{c}{z}e_3$$ Then we have $T(x,y,z)= (a,b,c)$. That is rotation matrix is matrix of $T$ that is $$\left( \begin{array}{ccc} \frac{a}{x } &0 & 0 \\ 0 &\frac{b}{y} &0 \\ 0 & 0 &\frac{c}{z} \\ \end{array} \right)$$

If some of x,y,z is zero, then case reduces to $\mathbb R^2$ keeping one axis fixed.

Example: Assume $u= (1,2,3)$ and $v= (2,-5,6)$ Then rotation matrix which take $u$ to $v$ is $\left( \begin{array}{ccc} \frac{2}{1} &0 & 0 \\ 0 &\frac{-5}{2} &0 \\ 0 & 0 &\frac{6}{3}\\ \end{array} \right)= $ $\left( \begin{array}{ccc} 2 &0 & 0 \\ 0 &\frac{-5}{2} &0 \\ 0 & 0 &2 \\ \end{array} \right)$

putting value of $a,b,c$ and $x,y,z $ you may have many more example.... If some of $x,y,z$ is what happen see and if didn't get comment it.. i will give example for that too.

Answer 2

I believe this article can help: Arbitrary Axis Rotation

Answer 3

Two 3-vectors define a plane. Rotation in that plane (i.e. about normal vector of that plane) brings one vector to another. So the quick sketch for the solution would be:

1. find normal vector to the common plane (I think this is just the vector product $u \times v$)
2. find rotation angle $\theta$ using dot product ($\theta = \cos^{-1}(\frac{u \cdot v}{||u|| ||v||})$)
3. express the rotation using some axis-angle representation (axis is the normal vector from 1. and angle is the $\theta$ from 2.)

I am curious if someone expresses the straightforward formula here...