Right-Handed Rule

I don't understand the role of the right-handed rule in the equation $$\omega = \frac{\theta - \theta_0}{\Delta t}$$. Could someone please explain it to me? I searched for some references, but they seem to talk about electricity instead. Also, does the right-handed rule have any other uses in rotational kinematics besides the angular velocity formula I posted?

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The direction of a cross-product pseudo-vector is determined by the right-hand rule. The direction your thumbs curl determine the direction of the cross product pseudo vector.

A counter-clockwise rotation (Or counter-clockwise cross product, if you can follow my meaning) would be produce an angular velocity vector pointing up, perpendicular to the plane of the rotation.

What you forgot in your definition of angular velocity, is that it is a vector.

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An additional explanation for you to ponder about why there exists a right hand rule. Imagine picking a point in space. Now draw a line out from that point in some direction. Lets decide to call this line the positive x-axis. Now lets decide to draw another line from this point such that it is at 90 degrees to the first line. Lets call this positive y-axis. Now these two lines can be imagined to exist inside a plane. Now we want to add a third dimension by drawing a third line out from the point that is 90 degrees to the other two lines, or 90 degrees to the plane. But we have two choices to do so, there is a "top" side of the plane and a "bottom" side. How do we make sure that everyone doing math problems with this coordinate frame gets the same answer? We make a definition. We decide to use our right hand to define a specific direction so that everyone following the rule gets the same answer. We decide that you must curl your right fingers from the x-axis to the y-axis and that whatever direction the thumb is pointing in will be the positive z-axis. We could have made the definition using the left hand, there is nothing magic about the right hand. As long as everyone agreed on the left hand, math and physics would have still worked out just the same.

The same argument works with two vectors. Connect vector tail of a vector A to the tail of a vector B. These two vectors define a plane. If want to define a cross product operation that makes the C vector, C = A X B, point at 90 degrees to both A and B, or 90 degrees to the plane, we have two choices. The right hand rule makes sure everyone picks that same choice.

So the right hand rule will come in where any cross products are used. In mechanics, there is Torque, Angular Momentum, Angular velocity, etc. In electrodynamics, there are cross products used as well. Any math or science problem that involves a cross product of some kind will require the use of the right hand rule.

Are you confused about the right-hand rule specifically to that equation? or just using it in general? I was taught differently than how RoyalCat explained it, although that works well too.