title: Finding rotation matrix to transform one (3-vector) basis to another in 3d
body: <p>We have basis in 3d, <code>Vx</code> = (1,0,0), <code>Vy</code> = (0,1,0), <code>Vz</code>=(0,0,1), which is transforming by (rotation) matrix M to <code>Vx'</code>, <code>Vy'</code>, <code>Vz'</code> respectively. So we have 3 equations:</p>
  <pre><code>M * Vx = Vx'
  M * Vy = Vy'
  M * Vz = Vz'
  </code></pre>
  <p>Thus we have 9 linear equations for 9 components of matrix M.
  Now I need to transform this equation in form <code>A * m = b</code> (to solve it with numpy i.e.), where m is column-vectors of unknown <code>M</code> components like <code>[m11, m12, m13, m21, ...]</code>, <code>A</code> is coefficients matrix, <code>b</code> is coefficients column-vector.</p>
  <p>So the question is, what are formulas for <code>A</code> and <code>b</code> ? Is it possible to write some matrix formulas for it?</p>
  <p>=== UPDATE</p>
  <p>I got 'Notable question' prize for this question so I am updating it to unblock. Please note the question is quite precise and I am not searching any tool, the phrase 'Or, is there any tool which will help to write per-component formulas?' is just small addition which is not matter a lot I think.</p>