somehow associated with a certain plane, and it is just because there are three dimensions in space that we can associate the quantity with a direction perpendicu lar to that plane. Of the two possible directions, we chose the"right-hand direction So if the laws of physics are symmetrical, we should find that if some demon were to sneak into all the physics laboratories and replace the word"right "for "left"in every book in which"right-hand rules"are given, and instead we were to se all "left-hand rules, "uniformly, then it should make no difference whatever Let us give an illustration. There are two kinds of vectors. There are"honest vectors, for example a step Ar in space. If in our apparatus there is a piece here and something else there, then in a mirror apparatus there will be the image piece and the image something else, and if we draw a vector from the piece"to the"some thing else, "one vector is the mirror image of the other(Fig. 52-2). The vector arrow changes its head, just as the whole space turns inside out; such a vector we call a But the other kind of vector, which has to do with rotations, is of a different nature. For example, suppose that in three dimensions something is rotating as shown in Fig. 52-3. Then if we look at it in a mirror, it will be rotating as indicated namely, as the mirror image of the original rotation. Now we have agreed to repre- mirror image, A step in space and its Fig.52-2 sent the mirror rotation by the same rule, it is a"vector"which, on reflection, does not change about as the polar vector does, but is reversed relative to the polar rectors and to the geometry of the space; such a vector is called an axial vector. Now if the law of reflection symmetry is right in physics, then it must be true that the equations must be so designed that if we change what corresponds to reflection, nothing will happen. For instance, when we write a formula which says that the angular momentum is L =r X p, that equation is all right, because if ye change to a left-hand coordinate system, we change the sign of L, but p and r do not change; the cross-product sign is changed, since we must change from a right-hand rule to a left- hand rule. as another example, we know that the force mirror image. Nole a Fig. 52-3. A rotati right-to a left-handed system, since F and v are known to be polar vectors the sign direction vector"isthat on a charge moving in a magnetic field is F v xB, but if we change from a velocity vector"is not reversed in angular change required by the cross-product must be cancelled by a sign change in B, which means that B must be an axial vector. In other words . if we make such a reflection, B must go to-B. So if we change our coordinates from right to left, we must also change the poles of magnets from north to south Let us see how that works in an example. Suppose that we have two magnets as in Fig. 52-4. One is a magnet with the coils going around a certain way, and with current in a given direction. The other magnet looks like the reflection of the first magnet in a mirror--the coil will wind the other way, everything that happens inside the coil is exactly reversed, and the current goes as shown. Now, from the laws for the production of magnetic fields, which we do not know yet officially, but which we most likely learned in high school, it turns out that the magnetic field is as shown in the figure. In one case the pole is a south magnetic pole, while in the other magnet the current is going the other way and the magnetic field is reversed--it is a north magnetic pole. So we see that when we go from right Fig. 52-4 to left we must indeed change from north to south! net and its mirror Never mind changing north to south; these too are mere conventions. Let us talk about phenomena. St have an electron movin one field, going into the page. Then, if we use the formula for the force, B (remember the charge is minus), we find that the electron will deviate in the indi cated direction according to the physical law. So the phenomenon is that we have a coil with a current going in a specified sense and an electron curves in a certai way-that is the physics-never mind how we label everything Now let us do the same experiment with a mirror we send an electron through in a corresponding direction and now the force is reversed, if we calculate it from the same rule, and that is very good because the corresponding motions are then mirror images