Why do we not define vector multiplication and division? Consider, two planar vectors:
$$V=  a \hat{x} + b \hat{y}$$
And
$$ U = a' \hat{x} + b' \hat{y}$$
These are analogous to the complex numbers:
$$ v = a + bi$$
and,
$$u= a' + b' i$$
Now, there are clear rules for multiply $ v \cdot u$ and also the expression: $ \frac{v}{u}$ , also there exists a geometric interpretation (if you view in polar form). Then, why is it that we don't bring up these products when speaking vectors and only discuss dot and cross product?
 A: Your proposed product is geometrical in a sense, but there is a crucial difference between it and the dot and cross products. Imagine you have a table, or some other flat horizontal surface, and two arrows drawn on it that represent vectors. Here is a question: What is the "complex number" product of these two vectors?
It is not possible to answer this question. This is perhaps most easily seen by noting that you cannot find the polar form of the vectors, because the angle depends on a coordinate system ("basis"), and we haven't specified one. In other words, the "complex number" product depends not only on the vectors, but on an (entirely arbitrary) choice of coordinates.
By contrast, the dot product and cross product can be understood without reference to any particular coordinate system. The dot product is the same in any orthonormal basis (that is, a coordinate system made from vectors of unit length that are all perpendicular to each other). This is also true of the cross product, provided the two bases have the same handedness (that is, you can rotate one basis into the other).
A: For planar vectors we can indeed multiply and divide.
First identify $V = a \hat{x} + b \hat{y}$ with $(a, b)$ and
$U = c \hat{x} + d \hat{y} $ with $(c, d)$
Now define $(a, b) \times (c, d) := (ac - bd, ad + bc)$.  You can check that this definition of multiplication is well-defined.
Also, $\frac{(a,b)}{(c,d)} = \frac{1}{c^2 + d^2}(ac + bd, ad - bc)$ is well defined.
These formulas may look foreign at first site but a closer inspection reveals that they are just multiplication and division of complex numbers.
