I was reading up on how to find the square root of i , and I learned that multiplication of complex numbers could be viewed geometrically by viewing the complex numbers as coordinates on the complex number plane $a_1+b_1i = (a_1,b_1)$ and $a_2+b_2i = (a_2,b_2)$. One can take the polar coordinates of the complex numbers to give $(a_1,b_1) \Rightarrow r_1$, angle = $w$ and $(a_2,b_2) \Rightarrow r_2$, angle = $k$ . And finally the multiplication of the two numbers can be viewed as multiplying $r_1$ and $r_2$, while adding the angles $w$ and $k$, to give the product $r_3= (r_1)(r_2)$, angle = $w+k$ . This can be used to intuitively find the square root of $0 + 1i$.
However in my math textbooks I have not seen any type of vector multiplication similar to this in regards to the real numbers, only dot products and cross products. Does this type of multiplication serve some purpose in regards to real number vectors, does it describe something interesting? Or is it only useful when it comes to multiplying vectors in the complex number plane?
Edit: $r$ is describing the overall length, or magnitude, of the vector. The angle represents the direction the vector is pointing in in regards to the plane it's on.