Many students have asked me why is there a difference between cross(vector) and dot(scalar) product, after all it is like multiplying. For starters, the former produces a resultant vector while the latter produces a scale quantity, as its name suggests. So is there an intuitive explanation?
If you know a bit of physics, it will be quite easy. Simply consider electromagnetic induction, there is a resultant work done and a resultant force. The resultant work done is our scalar product here while the resultant force (Fleming’s left hand rule) is our vector product here. I hope this helps your understanding!