But either way, the vector's components stay the same — and therefore the arrow, and the basic rules for multiplying vectors, remain the same, too. Hamilton defined two ways to multiply vectors ...

Hamilton defined two ways to multiply vectors together. One produces a number (this is today called the scalar or dot product), and the other produces a vector (known as the vector or cross product).

But that is multiplication for scalar variables. How do you multiply vectors? There are two common operations you can do with vectors. First is the dot-product (also known as the scalar product).