That result is clear, for if we only turn our axes in the plane, the twist around z in that plane is no different than it was before, because it is the same plane! What will be more interesting is the expression for Tyy because that is a new plane We now do exactly the same thing with the y' z -plane, and it comes out as follows Ty'?'=(cos 8-x sin O)Fz z(F Fr sin 8) =(yF-zFy)cos 8+(aFx -xFi)sin 8 Ty cos 8 Tzr sin 8. Finally, we do it for zx: T2r=z(Fr cos 6+ Fy sin 8) (x cos 0+ y sin O)Fz (zFx- xFz)cos 8-, -, =T cos 6-Tuz sin 8 We wanted to get a rule for finding torques in new axes in terms of torques in old axes, and now we have the rule. How can we ever remember that rule If we look carefully at(20.5),(20.6), and (20.7), we see that there is a close relation ship between these equations and the equations for x, y, and z. If, somehow, we could call Try the z-component of something, let us call it the z-component of T, then it would be all right; we would understand (20.5)as a vector transformation since the z-component would be unchanged, as it should be. Likewise, if we associate with the yz-plane the x-component of our newly invected vector, and with the zx-plane, the y-component, then these transformation expressions would Tr cOS , Ty=Ty cos 8-T sin 8, hich is just the rule for vectors Therefore we have proved that we may identify the combination of xFy - yFa with what we ordinarily call the z-component of a certain artificially invented vector. Although a torque is a twist on a plane, and it has no a priori vector char the mathematically it does behave like a vector. This vector is at right angles to plane of the twist, and its length is proportional to the strength of the twist. The three components of such a quantity will transform like a real vector So we represent torques by vectors; with each plane on which the torque is supposed to be acting, we associate a line at right angles, by a rule. But"at right angles"leaves the sign unspecified. To get the sign right, we must adopt a rule which will tell us that if the torque were in a certain sense on the xy-plane, then the axis that we want to associate with it is in the"up"z-direction. That is, some- body has to define"right"and"left"for us. Supposing that the coordinate system is x, y, z in a right-hand system, then the rule will be the following: if we think of the twist as if we were turning a screw having a right-hand thread, then the direction of the vector that we will associate with that twist is in the direction that the screw ould advance Why is torque a vector? It is a miracle of good luck that we can associate a single axis with a plane and therefore that we can associate a vector with the torque; it is a special property of three-dimensional In two dimensions. the torque is an ordinary scalar, and there need be no direction associated with it. In three dimensions, it is a vector. If we had four dimensions, we would be in great difficulty, because(if we had time, for example, as the fourth dimension)we would not only have planes like xy, yz, and zx, we would also have tx-, ty, and tz-pla anes There would be six of them, and one cannot represent six quantities as one vector in four dime We will be living in three dimensions for a long time, so it is well to notice that the foregoing mathematical treatment did not depend the fact that x