Vectorial Analogue To Usual Multiplication of Complex Numbers

Question

I know that a complex number can be taken as 2-dimensional vector and the dot and cross products have also been defined for two complex numbers but different from those of vectors.

But my questions is "What is the vectorial analogue to the usual multiplication of complex numbers? Also the product of two complex numbers is again a complex number"

I think "when it comes to the usual multiplication of complex numbers then complex numbers can't be treated like vectors anymore".

I mean that can we take two vectors and define their multiplication the same way like (a, b)(c, d)= (ac-bd, ad+bc) where ai+bj and ci+dj are two vectors? Please correct me if I am wrong providing a sound exposition.

Answer 1

When you multiply

$$ z_1 = r_1 e^{i\theta _1} $$

and

$$ z_2 = r_2 e^{i\theta _2}$$ you get

$$z_1z_2 = r_1r_2 e^{i (\theta _1+\theta _2)}$$

That is the norm of the product is the product of the norms and the argument of the product is the sum of arguments.

Thus if you view complex numbers as vectors, the multiplication is a composition of a rotation and a dilation.

Answer 2

I take your question to be whether one can define direct products of vectors just like we have a direct multiplication of vectors in the complex field.

Indeed we can, and that is what is called the exterior product of the vectors in the space. It enjoys some interesting properties too and leads to many important results. Any vector space with such a direct product of vectors is known as an algebra.