Traditionally, multiplication takes two elements of a set and turns them into another element of that set. The dot product doesn't do that, it takes two vectors and turns them into a scalar.
The cross product only exists in three dimensions. And it's not associative (A×B×C gives a different answer depending which order you do it in), which is another thing multiplication usually satisfies.
There are two other not-quite-multiplication operators that I recall seeing. There's an analog of the cross product in two dimensions: (a,b,0)×(c,d,0) = (0,0,ad-bc), so it can be useful to have an operator (a,b)×(c,d) = ad-bc, again turning two vectors into a scalar.
And if the dot product is defined in terms of matrix multiplication by A·B = AᵀB, then you can also define an operator ABᵀ, turning two vectors into a matrix. These vectors don't even need to have the same length.
> The cross product only exists in three dimensions
While true, the Wedge product is a useful concept that generalizes a cross product to arbitrary dimensions and is used in multivariable calculus for proving various integral theorems in high dimension. Here, generalized Stokes theorems apply despite the cross product not being defined. Admittedly, it isn't really a map on the vector space, but the fact that Stokes theorems still hold makes it pretty darn useful to me.
The cross product only exists in three dimensions. And it's not associative (A×B×C gives a different answer depending which order you do it in), which is another thing multiplication usually satisfies.
There are two other not-quite-multiplication operators that I recall seeing. There's an analog of the cross product in two dimensions: (a,b,0)×(c,d,0) = (0,0,ad-bc), so it can be useful to have an operator (a,b)×(c,d) = ad-bc, again turning two vectors into a scalar.
And if the dot product is defined in terms of matrix multiplication by A·B = AᵀB, then you can also define an operator ABᵀ, turning two vectors into a matrix. These vectors don't even need to have the same length.