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

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...

Answer 2

Quaternions are what you are looking for, don't let the word scare you it is very easy, here is matlab code to do what you want. (Quaternions allow for easy rotations about any arbitrary axis)

% example vectors
u = [1/sqrt(3),1/sqrt(3),1/sqrt(3)]; % vector to rotate
v = [0 0 1]; % vector to align to

% angle between vectors
theta = acos(dot(u,v)/(norm(u)*norm(v)));

% find axis perpendicular to u and v (cross product) to rotate about
r = cross(u,v);
% rotation axis must be a unit vector
r = r/norm(r);

% quaternion to rotate [0,ux,uy,uz]
q1 = [0,u];
% quaternion rotation matrix, rotates angle theta about axis r
q2 = [cos(theta/2),r(1)*sin(theta/2),r(2)*sin(theta/2),r(3)*sin(theta/2)];
q3 = quatmultiply(q2,q1);
q3 = quatmultiply(q3,quatconj(q2));

u_new = q3(2:4);

% plot the vectors
figure(1)
clf
hold on
plot3([0,u(1)],[0,u(2)],[0,u(3)],'b')
plot3([0,v(1)],[0,v(2)],[0,v(3)],'k')
plot3([0,u_new(1)],[0,u_new(2)],[0,u_new(3)],'r--')
axis equal
grid on

Answer 3

I believe this article can help: Arbitrary Axis Rotation

Answer 4

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.