Let's consider all of these results together.

In order to compute the matrix, V we need only compute the inverse of the matrix formed by concatenating our 3 special vectors. We will instead employ a little trick from linear algebra.

Remember that each of our vectors are unit length (we normalized them). Also, each vector is perpendicular to the other two. These two conditions on a matrix makes it, orthogonal. Rotations are also orthogonal. Orthonormal matrices have the unique property that: