If you have heard of cross product before, then you may have heard the term right-hand (RH) rule as well. If not, no worries! Take your right hand and stick your pointer finger out. Then rotate your middle finger to make a 90° angle with your pointer finger. Lastly, point your thumb up in the air.
Your pointer finger represents the x-axis, your middle finger represents the y-axis, and your thumb represents the z-axis. An example of this is shown in Figure 1.
Fig. 1 - Image of RH Coordinate System
Another way I like to think of RH coordinate systems is for cross products. If you had two vectors, $ X and Y $, and you were solving for $ Z $, where $ Z = X \times Y $, start by pointing your fingers in the direction of the $ X $ vector. Then, curl your fingers towards the $ Y $ vector taking the shortest path. The direction your thumb is pointing is the direction of the $ Z $ vector!
This is incredibly useful for vector analysis and shows up everywhere, such as in magnetic field and electron flow analysis.
We can continue this example by looking at a typical x-y-z coordinate system.
Fig. 2 - XYZ Coordinate System
Using Figure 2, we can find that
$$ X \times Y = Z $$
$$ Y \times Z = X $$
$$ Z \times X = Y $$
We can also flip the order of the vectors in the cross product to get the negatives of the output.
$$ Y \times X = -Z $$
$$ Z \times Y = -X $$
$$ X \times Z = -Y $$
The following 3 vectors above are often referred to as the left-hand (LH) rule, where
$$ Y \times X = Z $$
$$ Z \times Y = X $$
$$ X \times Z = Y $$
The LH Rule is essentially the opposite of the RH Rule! These rules are very common in physics and engineering and are, hence, incredibly useful to learn and start utilizing.