Given two unit vectors what can we say about the rotations that can transform one into another?

Question

Given 2 unit vectors in 2 dimensions there is a uniqe rotation matrix that transforms on to the other. $A$ and $B$ are the two unit vectors is there a general way to find the set of orthogonal rotation matrices that rotate $A$ to $B$?

Answer 1

$\begin{bmatrix} \cos\theta &-\sin\theta\\\sin\theta&\cos\theta \end{bmatrix}$ is the standard rotation matrix.

What is $\theta$?

$a\cdot b = \cos\theta$ (if $a,b$ are unit vectors)

But what direction?

$a_1b_2 - a_2b_1 > 0$ then $b$ is counter-clockwise from $a$ And if it is less than 0, then $b$ is clockwise from $a.$

You might think of the cross product, make these into 3 dimensional vectors with $\bf k$ term $0$ for both vectors, $a_1b_2 - a_2b_1$ will be the $\bf k$ term in the cross product.

Answer 2

Here is a geometric procedure

We will work on the surface of the unit sphere. The 2 unit vectors given are points p,q fixed on this sphere. Now take any other (variable) point x on this surface. These 3 points (not being collinear) lead to a unique circle C passing through them. That circle will lie on this surface, call its centre (not on the surface) O.

Now consider the diameter L of the sphere passing through O. Using L as axis there is a unique rotation sending p to q.

Now the third point x was arbitrary;varying it gives different choice of circle C and different diametric line L and a different rotation. sending p to q.