Get the unit vectors parallel to the cables using the coordinates of the end points. Express the tensions in terms of these unit vectors, and solve the equilibrium conditions. The coordinates of points A, B, C, D, O are found from the problem sketch: The coordinates of the points are A(6, 2, 0), B(12, 3, 0.), C(0, 8, 5), D(0, 4, -5), O(0, 0, 0).
The unit vector parallel to the tension acting between the points A, B in the direction of B is by definition
Perform this for each of the unit vectors
The tensions in the cables are expressed in terms of the unit vectors,
The equilibrium conditions are
Substitute and collect like terms,
This set of simultaneous equations in the unknown forces may be solved using any of several standard algorithms. The results are: