I do not understand how does the result of two vectors acting on a particle require me to take the cross product to find the resultant.
Won't the actual force on the particle be the result of the vectorial addition of the two vectors? How is it that result of two vectors on a charged particle in a magnetic field (the magnetic field vector and the velocity vector of the particle) has the resultant force on the particle in a mutually perpendicular direction?
I'm not sure if I understood what you mean.
You want to know why the Lorentz force is given by, $$ \vec F = q \vec v \times \vec B $$ instead of $$ \vec F = q\vec v + \vec B\,, $$ am I right? For a start, the second equation is not dimensionally correct. It would if $\vec B$ would be a generic outer force and $q$ would be some fluid friction coefficient, but neither of those quantities are such. $q$ is a charge, $\vec v$ is a velocity and $\vec B$ is a magnetic field. They are not forces, and should be not treated as forces.
This is the reason why the second equation is wrong. As for the reason why the first equation is right, the best answer i can give you is just "because". It is just a fact, an axiom you base electromagnetism on. You could reprhase it in term of other axioms regarding the simmetry of reality between electric forces and magnetic forces, using field theoretical arguments (e.g. introducing the electromagnetic tensor $F_{\mu\nu}$ and its coupling to the 4-current $J_\mu$), but there is no specific reason (as far as we know) by which reality should have these symmetries rather than others.
The magnetic field is not a vector. It is a pseudovector, it transforms differently upon space inversion (mirror reflection if you prefer). The most natural way to construct a vector from a vector and pseudo vector is to take their cross product.
