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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 $v$?

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No need to "rotate": divide $v$ by its norm, then multiply by the norm of $u$. –  David Mitra Feb 27 '12 at 16:57
    
@David: I don't understand; scaling doesn't change whether $u$ and $v$ are linearly dependent. –  Zev Chonoles Feb 27 '12 at 16:59
    
@ZevChonoles Maybe I misunderstood the question. I thought he ultimately wanted a vector in the direction of $v$ that had length $\Vert u\Vert$. –  David Mitra Feb 27 '12 at 17:02
    
But if $u$ is not initially in the same direction as $v$, scaling won't change that. –  Zev Chonoles Feb 27 '12 at 17:07
    
@David, to clarify, $u$ and $v$ are not initially pointing in the same direction. I'd like to rotate $u$ so that it points in the same direction as $v$, not rescale it. –  FlyWheel Feb 27 '12 at 17:08
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3 Answers

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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.

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How do we find the matrix for the linear transformation $T$? –  FlyWheel Feb 27 '12 at 17:25
    
@FlyWheel, read this article en.wikipedia.org/wiki/Transformation_matrix –  zapkm Feb 27 '12 at 17:27
    
@FlyWheel math.stackexchange.com/questions/92206/… This question may help you in leaning find matrix of linear transformation.. –  zapkm Feb 27 '12 at 17:29
    
I really appreciate you taking the time to answer my question. But if you could spare a moment, could you give me a simple example of rotating $u$ to $v$ in 3-space? –  FlyWheel Feb 27 '12 at 17:32
    
@FlyWheel, I edited the answer as your requirement... –  zapkm Feb 27 '12 at 17:41
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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...

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I believe this article can help: Arbitrary Axis Rotation

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Thanks, that's a nice link. However, my question is about how one actually comes up with a set of rotations so that $u$ can point in the same direction as $v$. –  FlyWheel Feb 27 '12 at 17:09
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