Suppose there are three orthogonal unit vectors, A, B, C, where each unit vector is perpendicular to a cartesian 3D co-ordinate system X, Y, Z. Each unit vector is defined by the vector connecting the X, Y, Z origin to a point in 3D space. In this model:

A = (1, 0, 0) lies along the X axis = (Ax Ay Az)
B = (0, 1, 0) lies along the Y axis = (Bx By Bz)
C = (0, 0, 1) lies along the Z axis = (Cx Cy Cz)

The unit vectors are then manipulated such that they are now represented by:

A' = (A'x A'y A'z)
B' = (B'x B'y B'z)
C' = (C'x C'y C'z)

The lengths are still unit e.g. sqrt((A'x)(A'x) + (A'y)(A'y) + (A'z)(A'z)) = 1. The dot products can be calculated to prove that A', B' and C' are still orthogonal.

Now to the problem. There is a cuboid centered about the X, Y, Z origin represented by 8 points in 3D space. Say that each point on the cuboid can be represented as (u, v, w) such that the 8 points are:

(u, v, w) (-u, v, w) (u, -v, w) (-u, -v, w)
(u, v, -w) (-u, v, -w) (u, -v, -w) (-u, -v, -w)

The axes of this cuboid can be considered to be parallel with the X, Y, Z axes. The question is how to transform the cuboid in 3D space such that the axes are made parallel to the A', B', C' unit vectors?