Multiplying two scalars and means starting with a total of zero and adding to the total times or equivalently of adding to the total times. Geometrically, we interpret multiplication as an area. Linguistically, multiplication often shows up as the word "of". Scalars just have size, and while there are many interpretations of multiplying two scalars together, there is only one mathematical process.
Vectors have both size and direction. Intuitively, we would like the operation of multiplication to reflect both of these characteristics of each vector. For direction we can think about both the absolute and relative directions of the two vectors.
In terms of components of vectors in a 3D cartesian space, we expect that the result of multiplication should be made up of terms which are products of one component from each vector. There are 9 of those. If think about all possible combinations of these 9 terms involve coefficients of -1, 0, or 1, there are nearly 20,000 possibilities. These factors can be combined into scalars, vectors, or tensors. Below, a few of those with nice symmetries are explored.
The vectors and , in orange, are to be multiplied. Their size can be controlled by the sliders in the text window. Their orientation can be controlled by moving point C to change the orientation of the plane they lie in and by moving points A and B around circle (defined by the intersection of a unit sphere with the plane whose normal is from the origin to C).