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I am well aware that a charged particle moving in a magnetic field will experience a force perpendicular to that magnetic field. But why is it that positive and negative particles experience a force in opposite directions?
What exactly determines the direction that a given charge will experience a force? I.e. why does a negative particle experience a force in one direction and not the other?
The problem with the why in your question is that it will give rise to another why in the explanation to explain why that explanation occurs and so on; giving rise to an infinite number of whys. But to kick things off, I'll try and give an initial explanation:
Charges deflect in a magnetic field dependent upon their charge sign because of:
We can move to a laboratory moving at the same instantaneous velocity as the charge where it now appears stationary. Any magnetic field here cannot affect the charge because it's not moving; leaving only an electric field, if any, that can affect it. This electric field will deflect that the charge in one of two ways depending upon its sign, which will also be seen in one of two ways in the original laboratory where the charge was moving.
Now we're left with another two questions which I can't give an answer to:
Electromagnetism is symmetric with respect to parity. That symmetry is broken by the convention we choose to use for defining the magnetic field vector. Aliens on another planet could define magnetic fields to point in the opposite direction compared to our definition. They would then use a left-handed rule $\textbf{F}=-q\textbf{v}\times\textbf{B}$ rather than our right-handed $\textbf{F}=q\textbf{v}\times\textbf{B}$. If you get in radio contact with these aliens and try to get them to tell you whether their definitions are the same as ours or opposite, you can't tell without some external reference point that tells them which hand you consider right.
What exactly determines the direction that a given charge will experience a force? I.e. why does a negative particle experience a force in one direction and not the other?
why is it that positive and negative particles experience a force in opposite directions?
You can express the rules in ways that don't refer to the magnetic field or its arbitrarily defined flippable direction. For example, parallel current-carrying wires attract each other if the currents are in the same direction. Such rules are independent of which charges you define as positive and which way you define the magnetic field.
When expressed in these ways that avoid the arbitrary conventions, these rules follow from special relativity. The classic presentation at the freshman physics level is in the textbook by Purcell.
I'm not really sure to have understood your question, however you have to consider the sign of the charge inside the formula: $$\mathbf{F_{Lor}}=q\mathbf{v}\times\mathbf{B}$$ and so if: $$q=|q|$$ $$\mathbf{F_{Lor+}}=|q|(\mathbf{v}\times\mathbf{B})$$ instead if: $$q=-|q|$$ $$\mathbf{F_{Lor-}}=-|q|(\mathbf{v}\times\mathbf{B})$$ How you can see: $$\mathbf{F_{Lor-}}=-\mathbf{F_{Lor+}}$$ The force acting on a postive and on a negative charge are opposite vectors.
I've been told I can't just link to another site so I will try to paraphrase the article I linked to. The magnetic force is perpendicular to the velocity of the particle it is acting on. That causes the direction of the particle to change and travel in a circular motion. https://cnx.org/contents/bZRPyVNP@2/Motion-of-a-Charged-Particle-i
