why are there only two methods of vector multiplication?

Question

Vector multiplication, there are two types one scalar or dot as $A\cdot B=\vert A \vert \vert B \vert \cos \theta$ and vector or cross $A\times B = \vert A \vert \vert B \vert \sin\theta$ in geometric form and there respective algebric or component form. But what property of vectors makes them work as ways of vector multiplication and other possible ways don't. Basically what is motivation behind the scalar product and vector product how and why they perfectly justify multiplication of vectors.

Answer 1

One motivation for these types of vector multiplication is that we can perform geometric calculations in three-dimensions through purely algebraic methods; for example, since the dot product (or scalar product) of $(1,2,3)$ and $(4,-5,2)$ is $4-10+6 = 0$, we know that they are perpendicular to each other since the sine of the angle between them must be zero. All we had to do was add and multiply coordinates to arrive at this answer. Similarly, the cross product (or vector product) of two vectors gives us a third vector perpendicular to the first two with a magnitude given by the area of the parallelogram spanned by them. In addition, they have nice symmetry properties, such as commutativity ($A\cdot B = B\cdot A$) or anticommutativity ($A\times B = -B\times A$) and distributivity.

As to why we use these and not others; if other such products had the properties of 1)connecting geometric properties with algebraic computations and 2)having nice symmetry properties to make them applicable in practice, we would almost certainly use them. However, this turns out not to be the case. In fact, it turns out the cross product as it is defined can $only$ exist in three-dimensions; there is no real analogue to the 1, 2, or $n > 3$-dimensional case. The dot product however, is can be defined in any dimension with the same properties as usual. It therefore is more `natural' perhaps, for vectors in any vector space (which can be seen by its generalization to an inner product (https://en.wikipedia.org/wiki/Inner_product_space)).