3D rotation matrix and direction cosines of the transformed coordinate system.

Question

I have an orthogonal basis coordinate system and a rotated coordinate system of which I know the orthonormal direction vectors $ (\overrightarrow{e}_1, \overrightarrow{e}_2, \overrightarrow{e}_3) $ with respect to the basis system. In a reference I found that the rotation matrix consists then of the components of the direction cosines (written as columns). This seems contra-intuitive to me; is there a simple method to prove / disprove it?

Answer 1

The components of a unit vector wrt the basis system are its direction cosines.

The columns of a transformation matrix are the images of the basis vectors.

So this property works for any linear transformation, provided you normalize the column vectors.