This description always bothered me. You have some space (like the actual space you’re in), and you have three vectors (which could be very real, e.g. the velo

This description always bothered me. You have some space (like the actual space you’re in), and you have three vectors (which could be very real, e.g. the velocities of three balls, labeled A, B, and C, and you calculate this particular function of the vectors (velocities), and you get a number, say 5. And then your professor tells you that 5 is a pseudoscalar, and if you change coordinates, it’s actually -5. And you boggle, because there weren’t any coordinates to begin with — there were just actual balls, and either the answer is 5 or -5 or ±5 (like sqrt(25), for example), but it’s weird for the answer to be “5 or -5 depending on coordinates”. And then people struggle on the tests, because they didn’t get it, because the material was taught poorly. What’s actually happening here is that you need an orientation to calculate the scalar triple product. If you have three moving balls and a right hand or a right shoe or something similar, you get 5. If you look in a mirror and calculate the triple product of the mirrored balls’ velocities, you get -5 and you see a left hand and a left shoe. But if you watch yourself in the mirror calculating the triple product from the balls, the mirror copy of yourself gets 5 correctly, because mirrored right shoes fit on mirrored right feet, and mirror-you is looking at mirror-balls, and everything is consistent. So a pseudoscalar is a quantity that gets negated in a mirror if the person calculating it isn’t also in a mirror. No coordinate systems needed. As a simple example, hold your hand in your favorite right hand rule gesture. Your thumb, index finger, and middle finger point along vectors, and their triple product will be positive for one hand and negative for the other hand. Now look in a mirror!