Why can the maximum torque be obtained for a brushless DC motor when its stator magnetic field is controlled to be orthogonal to its rotor magnetic field? What theory or what law can explain this phenomenon?

The torque is produced by the fact that the 2 fields (the one produced by the magnets on the rotor and the one produced by the stator currents) will tend to align to each other. This means that zero torque is produced when the fields are aligned, and torque production increases as they are "shifted" with respect to each other, reaching the maximum amplitude exactly at the orthogonal relative position, when they are at the furthest distance.
Mathematically, this is expressed by the equation:
T=k*F_stat*F_rot*sin(theta)
where F_stat and F_rot are the amplitude of stator and rotor magnetic field, respectively, and theta their angular displacement.

Francesco,
As always, a perfect explanation!
Accurate and to the point.
Nir

Francesco,
Thank you for the answer. In fact, I know the formula T=k*F_stat*F_rot*sin(theta), but I don't know how to derive it. From Faraday's law, Coulomb's law, Ampère's force law or some one else? I would like to know what theory or what law can support or explain this natural phenomenon.

If you look into lecture above, you will find F=I x B.
This is so called Lorentz Force, the force experienced by current cunduction wire in magnetic field.

I see how to explain this natural phenomenon. First, Ampère's force law F = I × B derived from Lorentz force F = qv x B tells us the direction and magnitude of the force acting on the stator coils F_rot2stat. Second, the stator coils is immovable so a force F_stat2rot equal in magnitude and opposite in direction is exerted to the rotor (Newton's 3rd law of motion). Third, Ampère's circuital law tells us the direction of stator magnetic field B_stat produced by the current in stator coils. Fourth, because Torque T = r x F_stat2rot, maximum torque is obtained when F_stat2rot is orthogonal to r. At this moment, B_rot and B_stat align to r and F_stat2rot, respectively. Therefore, the maximum torque can be obtained for a brushless DC motor when its stator magnetic field is controlled to be orthogonal to its rotor magnetic field.
Thanks Michele for providing the lecture, and thanks Jongsoo for the advanced answer.