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In addition to the well known right hand rule, there is indeed a left hand rule. This rule can also illustrate cross products like the Lorentz force law, but in a different manner. When using the right hand rule, you are using a right handed coordinate system and are assuming that all of the quantities are positive. Your thumb, index finger, and middle finger represent the X, Y, and Z positive axes, so the right hand rule works if all quantities are positive.

What if you are asked to find how an electron or other negative charge, rather than a positive charge, is affected by a magnetic field? Put both your hands in the position that you use for the right hand rule. Your thumbs and index fingers will both point in the same direction, but your middle fingers will be pointing in opposite directions. This is what would happen if you reversed one of the axes on the XYZ axis system. Reversing one of these axes would be like representing a negative quantity on that axis. Thus, you can use the left hand rule exactly as you would the right hand rule, except in different situations. Whenever you have one quantity that is negative, you can use the left hand rule and save yourself a lot of work.

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