Different vector "product" and its geometric significance

Question

Is there a name for the following vector "product" (let's symbolize it by op): $(a,b)$ op $(c,d)=(ac-bd,bc+ad)$? Does it have any geometric significance?

Answer 1

This is perhaps the most important product of plane vectors.

The geometric significance is that the resulting vector has length which is the product of the lengths of the two original vectors, and the angle it makes with the positive $x$-axis is the sum of the angles that the to original vectors made.

It is known as the product of complex numbers.

Answer 2

That's actually the product as defined on the complex numbers.

It can be seen as the application of a rotation of angle equal to the argument of $(c,d)$ and a scaling by a factor equal to the modulus of $(c,d)$.

When applied to $(a,b)$ and $(c,-d)$ (the conjugate of $(c,d)$) you have a vector composed of the dot and cross-products.