/sci/ - Science & Math

I am not a math major, so I may lack the correct terminology to describe what I mean:
I am trying to develop a 2-dimensional chart, where one axis (let's say x) does not detail something "spectacular", you can just assume it to model the reals. The other axis, however, uses a kind of different dynamic. I want the two points y = 1 and y = -1 to detail a point that semantically entails "total difference from a y-value at y = 0". This same relation would apply for any further movement along this axis, meaning "y = 2" would describe "total difference from y = 1 (and also y = 0)", and so on. As such, something like y = 1.5 would detail "this point is half-different from 1", or even "half different from 37".

So the dynamic is: [math][Degree of similarity] = (y' - y) (mod) 1[/math]

This is obviously somehow related to trigonometry and periodicity. You can envision the concrete y-values as the number of peaks that have elapsed on a sine graph.
But I need it to be a Cartesian field. The reason this is important because under the system I am trying to develop, consider e.g. the two tuples (22, 3) and (22,7). While both describe something where there is a "total difference" within the y dimension from e.g. (22,1) -- but the two tuples also would NOT point to an identical thing; as such, the separation is important. Indeed, that is precisely why I keep those numbers around -- to give some means to distinguish.
MY QUESTION IS: is there even a math or notation concept where it wouldn't seem arbitrary for such a Cartesian chart to have x and y axes with different functionalities? As one axis just tracks the reals, but the other axis somehow denotes a periodic function.

>>14869003
The most important thing to remember is that when a and b are not scalars, they have a direction. Thus, there is an angle between a and b, and by swapping a and b, you are inverting the angle.
>>14869197
You don't need to understand those 20 lines of algebra. You simply need to understand that a cross product is:
[math]\mathbf{a} \times \mathbf{b} = \left\| \mathbf{a} \right\| \left\| \mathbf{b} \right\| \sin (\theta)[/math]
whereas a dot product is:
[math]\mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|\ \|\mathbf{b}\|\cos\theta[/math]
Pic related shows that inverting a nonzero angle flips the sign of the sine, but not of the cosine. Thus, the dot product--which depends upon the cosine--is commutative (but only for real numbers), while the cross product--which depends on the sine--is anticommutative.