Figure 1 Coordinate system

Vector product
The vector product of two vectors and , also called the cross product, is denoted by . The simplest definition to understand is that the cross product of and produces a vector of magnitude

In a direction normal (perpendicular) to the plane of and with the angle between and .

Fig 2.29

Where , we can write Eq. 10

We will work not only with the magnitude but we should find a direction of vectors products of two vectors. The direction of is perpendicular to the plane containing and with its orientation given by the right–hand rule. The direction is obtained by rotating into using the fingers of the right hand naturally curling (closing) from to with thumb pointing in the direction of new vector. This is illustrated in Fig. 2.29a. Similarly, we determine the direction of by rotating into as in Fig. 2.29b. The result is a vector that is opposite to the vector .
The vector product is anti–commutative:
The vector product of any vector with itself(angle between two vectors is zero) is zero, so

Relations among the unit vectors for vector products by using right-hand rule are

Figure 2.30

Next we express and in terms of their components and the corresponding unit vectors, and we expand the expression for the vector product

Thus the components of are given


The vector product can be abbreviated by the notation of the determinant:

Figure 2.31

Example 12: Vector A is directed in the positive x-direction and has a magnitude of 5 and vector B makes angle of 30⁰ with the positive x-direction and has a magnitude of 4. Find the vector product
Solution: Using Eq. 2.20 we can find the magnitude of the vector product

We can prove this answer by using Eq. 2.22. First we should find x and y - components of and .

The same result we get by using two approaches. The direction of the vector product of points in the z-direction.