Lorentz force vector

I’m trying to calculate a Lorentz vector in $$\mathbb{R}^3$$ with those vectors but I keep getting an integer as an answer:

There’s a velocity vector of $$(3,-2,1)$$ , a magnetic field vector of $$(1,2,-2)$$ and a charge $$q$$ of $$-3$$ . I read somewhere that the electric field vector is perpendicular to the magnetic field vector, but wouldn’t that mean the electric field vector is the zero vector? And when I use the formula from https://en.wikipedia.org/wiki/Lorentz_force, should I use the inner/dot product when multiplying the velocity and magnetic field vector?

2 Answers

I read somewhere that the electric field vector is perpendicular to the magnetic field vector

That, in general is not true. Think of a positive charge inside a solenoid. The electric field would be parallel to the magnetic field at points which lie on the line parallel to the magnetic field, containing the position of the charge.

wouldn’t that mean the electric field vector is the zero vector?

Not necessarily, but if the problem doesn't specify what the electric field is at that point, it is often assumed to be zero. (Unless the magnetic field is changing with time, in which case, an electric field would be induced in the volume. However, in this case, we are not given the magnetic field in a region, we are only given it's value at a point, so we are not left with much choice.)

should I use the inner/dot product when multiplying the velocity and magnetic field vector?

No, you should use the cross product (also sometimes called the outer product) while multiplying the velocity vector and the magnetic field. Using the dot product would yield a scalar, whereas the cross product gives a vector (which the force should be).

It is not necessary for the electric and magnetic fields to always be perpendicular. You must have read about this in the very special case of an EM field in vacuum. A simple counter example is to consider a bar magnet placed next to a point charge.

As to calculating the magnetic contribution to the Lorentz force vector, as mentioned in the Wikipedia link you need to use the cross product when multiplying the magnetic and velocity vectors. The electric force acting on the charge cannot be determined from any of the information given above. Most probably you are required to assume it to be zero.