Although the multiplication of one vector by another is not uniquely defined (cf. scalar multiplication, which is multiplication of a vector by a scalar), several types of useful vector products can be defined, as summarized in the following table.

Vector multiplication can also be defined for vectors taken three at a time, as summarized in the following table.

A number of vector quadruple products can also be defined.

More things to try:

Weisstein, Eric W. "Vector Multiplication." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorMultiplication.html