The Vector or Cross Product

In Fig. 1 are shown two vectors and , which make an angle measured in the plane of the two vectors. The vector product is defined as a vector along line normal to the plane of the vectors and , with such a direction that, if one looks in the direction , a clockwise turn will bring the direction of into that of . The rule for direction may also be stated by saying that the direction is that of the advance of a right-handed screw turned from to . The magnitude of the vector product is defined as . The magnitude is thus equal to twice the area of the triangle shown shaded in Fig. 1. Because of the form in which it is written the vector product is often called the cross product.

From the above definition it is seen that the vector product depends upon the order in which the vectors are taken. The vector product would have the same magnitude as , but would have the opposite direction so that i.e., vector products are not commutative.

Writing the vector product in the form of components, we have:

From the definition of the vector product, we have the following relations between the unit vectors : hence: