Motional Emf confusion
I believe what you are asking about is the mathematical vector expression that both your scalar expression and Fleming's Left Hand Rule comes from:

This would be the equation for the Lorentz Force. Of course you could also just say Maxwell's equations.

Basically ripped off Wikipedia:

\$ \overline{F} = q\overline{E}+q\overline{v} \times \overline{B}\$

\$ \overline{F} = \overline{I}\ell \times \overline{B} \$

The 
\$ q\overline{v} \times \overline{B}\$ term is the motional EMF that you are interested in.

The \$q\overline{E}\$ is just for the force exerted on an an electrons moving parallel to the component of an electric field.


  When one of them isn't, then there would be no emf


This is wrong. You just stated that if they are not perpindicular then the force is zero. I'm sure this isn't what you have in your mind but this is what you said.

What it is saying is that the mutually perpendicular components of all three terms determine each other. Not that they are zero if the vectors are not all perpendicular to one another.