We found a reference wherein the author follows the approach below to determine the directional cosines :

l1 , m1, n1 -----which are the cosines of the angles between x' axis and the global x, y and z axis

l2 , m2, n2 -----which are the cosines of the angles between y' axis and the global x, y and z axis

l3 , m3, n3 -----which are the cosines of the angles between z' axis and the global x, y and z axis

However, we have not understtod the appraoch.I will describe below what he has done.

l1 , m1 and n1 is clear; that is;

l1 = x2 - x1 / length (AB)

m1 = y2 - y1 / length (AB)

n1 = z2 - z1 / length (AB)

Let

Vx' be a matrix such that

[Vx' ] = [l1 m1 n1]^T

Then, he takes a reference point C such that C does not lie along the line joining AB

Let the coordinates of C be (x3,y3,z3)

He then gets:

"V13" where V13 is a matrix

V13 = [ (x3 - x1) / l13(y3 - x1) / l13(z3 - x1) / l13]

where,

l13 is the distance between A and the reference point.

Now,

[l3 m3 n3] ^T =(Vx ' x V13) / modulus of (Vx ' x V13)

Let:

[l3 m3 n3] ^T = Vz'

Note:(Vx ' x V13) denotes cross product

Next:

[l2 m2 n2] ^T = Vz' x Vx'

Note: (Vz ' x Vx') denotes cross product

We do not follow the approach above?

He has not given any explanation for above.

Whatever, he has given we have described above???