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

group-theoryorthogonal matricesrotations

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

Best Answer

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
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