The direction of the force on a current carrying wire in an external magnetic field is...?

Because the force on the individual moving charges has that property. F = qv x B where v and B are both vectors and the "x" indicates "cross product" which is perpendicular to both vectors.

Why is that true? We don't have a "why" answer to that more than why any other law of physics is the way it is in our universe. That's just the equation that describes the observed behavior correctly.

It is simply a fundamental observation of nature that the force a magnetic field exerts on a moving charge is perpendicular to both the field and the motion. But to elaborate a bit more on the mathematical interpretation of this...

What this means is that the speed of the moving charge will never change. In other words, no kinetic energy is added to a moving electric charge by a magnetic field, only an electric field can do that. So, the reason that it exhibits this behavior is because the magnetic field cannot do any work on an electric charge.

C. The drive is the go-product of the current vector and the B-field vector. This inspiration could be very effortless to internalize making use of the proper Hand Rule, where the thumb is the force, the index finger is the present, and the core finger is the B-subject. Hope that helps!