$3\times 3$ rotation matrix from axis of rotation and Angle

Question

I had encountered problem that tells to find $3\times 3$ rotation matrix from axis of rotation $(1,1,2)$ and Angle $\pi /3$.
I know that for axis of rotation on some standard vector like x axis, By putting $(1,0,0)$ in that first coloumn and then use rotation formula for $2\times 2$ case we can obtain required matrix.
On wikipedia I found one formula ,But there is no proof of that .
I wanted to know the creation process intuitively not just formula .
ANy help will be appreciated

Answer 1

HINT

A general way to find that matrix is as follow

1. Select an orthogonal basis $v_ 1,v_2,v_3$ with $v_3=(1,1,2)$
2. Consider the rotation matrix $M_B$ with respect to that basis (which is the standard rotation matrix around the $z$-axis)
3. Perform a change of basis from $v_ 1,v_2,v_3$ to the standard basis

Answer 2

Here is a method that I find intuitive.

For any given axis of rotation $(x_0,y_0,z_0)$ and any rotation angle $\theta,$ First we rotate by some rotation given by a matrix $R_1$ so that the vector $(x_0,y_0,z_0)$ is mapped to the positive $z$-axis. Then we rotate by angle $\theta$ around the $z$-axis; let $R_\theta$ be the matrix for this rotation. Then we undo the first rotation.

The rotation matrix around the axis $(x_0,y_0,z_0)$ then is given by the matrix product representing the rotations described above: $$ R = R_1^{-1} R_\theta R_1. $$

You have many choices for $R_1,$ because any "extra" rotation around the $x$-axis that $R_1$ might perform will be canceled by $R_1^{-1}.$ But one way is to rewrite your vector $(x_0,y_0,z_0)$ in spherical coordinates, $(\rho,\theta,\phi)$ where $\phi$ is the angle from the $z$-axis. To rotate $(x_0,y_0,z_0)$ to the positive $z$-axis, you rotate by $-\theta$ around the $z$ axis to map to a vector in the $x,z$ plane, then by $-\phi$ around the $y$ axis to map that vector onto the positive $z$ axis. Multiply the matrices of these two axis rotations in the correct sequence to get $R_1.$

It may help to do some graphing of a worked example in order to avoid sign errors.