What do you know about the planar product of two vectors?

[LCSphysicist]

TL;DR Summary

To summarize, i want to say i never saw something like this before, i am reading a book that defines a product of vectors with the properties: The A*B (A,B are vectors) product form a vector which lies in the plane of the vectors.

When i read this in the book "A VECTOR APPROACH TO OSCILLATIONS" i was a little shocked, because first it make quotients of vectors, and after this he defines this planar product, i searched this in google: i found nothing.

Anyway, this operations make sense if we imagine the vectors representing complex numbers, but yet, i don't think this is a general properties of vectors.

Do you know some reference about this?

 

[mfb]

You can define any operation you want as long as it's well-defined. This looks like a needlessly convoluted way to work with complex numbers without actually using complex numbers.

 

[nuuskur]

I read some of it and to me it looked like $A\angle{\alpha} \sim \|A\|\exp{(i \angle\alpha)}$. All the properties they discuss are that of the complex numbers.

This is known to the author as well, they point out the following.

Why they choose to go about it like this, I'm not sure.