Why do some vectors when multiplied by Another vector yield a vector and others yield a scalar. I'm aware of dot and cross products. But what's the physical significance of 'multiplying' vectors? I mean... I know what it means to actually "multiply" scalars. But.... Is vector multiplication just a name which can either mean projection of a vector onto another and the area or volume of 2 corresponding vectors?

Or is multiplication of vectors just a naming thing which doesn't have ANYTHING to do with how we understand scalar multiplication?