was position and F was force; it only depended on the transformation laws for vectors. Therefore if, instead of x, we used the x-component of some other vector it is not going to make any difference. In other words, if we were to calculate a,by-abx, where a and b are vectors, and call it the z-component of some new quantity c, then these new quantities form a vector c. We need a mathematical notation for the relationship of the new vector, with its three components, to the vectors a and b. The notation that has been devised for this is c a x b. We have then, in addition to the ordinary scalar product in the theory of vector analysis, a new kind of product, called the vector product. Thus, if c=ax b this is the same as writing (20.9) If we reverse the order of a and b, calling a, b and b, a, we would have the sign of c reversed, because c would be bzay-byaz. Therefore the cross product is unlike ordinary multiplication, where ab ba; for the cross product, b x a X b. From this, we can prove at once that if a= b, the cross product is The cross product is very important for representing the features of rotation and it is important that we understand the geometrical relationship of the three vectors a,b, and c. Of course the relationship in components is given in Eq. 20.9 and from that one can determine what the relationship is in geometry. The answer is, first, that the vector c is perpendicular to both a and b. (Try to calculate ca and see if it does not reduce to zero. Second, the magnitude of c turns out to be the magnitude of a times the magnitude of b times the sine of the angle between the two. In which direction does c point? Imagine that we turn a into b through an angle less than 1800; a screw with a right-hand thread turning in this way will advance in the direction of c. The fact that we say a right-hand screw instead of a left-hand screw is a convention, and is a perpetual reminder that if a and b are honest"vectors in the ordinary sense, the new kind of"vector"which we have created by a x b is artificial, or slightly different in its character from a and b because it was made up with a special rule. If a and b are called ordinary vectors, ve have a special name for them, we call them polar vectors. Examples of such vectors are the coordinate r, force F, momentum p, velocity v, electric field E, etc these are ordinary polar vectors. Vectors which involve just one cross product in their definition are called axial vectors or pseudovectors. Examples of pseudovectors are, of course, torque r and the angular momentum L. It also turns out that the angular velocity w is a pseudovector, as is the magnetic field B In order to complete the mathematical properties of vectors, we should know all the rules for their multiplication, using dot and cross products. In our applica tions at the moment, we will need very little of this, but for the sake of completeness we shall write down all of the rules for vector multiplication so that we can use he results later. These are (a)a×(b+c)=a×b+a×c, (b) (c)a·(b×c)=(a×b)·c, (d)a×(b×c)=b(a·c)-c(a·b), (e) a=0, (f)a·(a×b)=0. 20-2 The rotation equations using cross products Now let us ask whether any equations in physics can be written using the cross product. The answer, of course, is that a grea eat many equations can be so written. For instance, we see immediately that the torque is equal to the position