The Cross Product (prodotto esterno di vettori)The product of two vectors in the space according to the cross or vector product is a another vector. This new vector is perpendicular to the plane in which the first two vectors lie. Note that there are two possible directions in which the cross product of two vectors may point. This potential problem is solved by the right-hand rule. We will use the notation v x w - hence the name "cross product."
If v = w, then these "two" vectors are one, and there is no single plane determined by them. Thus, we require that v x v = 0.
We define the cross product for pairs of the basic unit vectors i, j, and k. Each of these is perpendicular to the plane of the other two, so we can define i x j to be k. The usual convention for coordinates in space is the right-hand rule, as illustrated in the following figure:
That is, if we follow the direction of the fingers to go from the x-axis to the y-axis, then the thumb points in the direction of the z-axis. The unit vectors i, j, and k point in these same directions. Thus, if we adopt the right-hand rule for cross products as well, then we want i x j to be k, as illustrated in the following picture: