The dot product discussed in the previous section, was introduced through the requirement that arose in calculating the work done by a given force when the point of application of the force is displaced by a certain amount given by :
In this section, we’ll see that another form of vector product exists and is extremely useful to discuss many different physical phenomena; this product is called the cross product. The cross product of and is another vector and the relation is represented as
Let us, through a physical example, understand what the cross product means.
Consider a horizontal magnetic field, which we can represent by , and a charge projected into this field with a velocity (at an angle with the horizontal).
Experiments show that the force acting on this particle
(a) is perpendicular to the plane of and and goes into the plane for the figure above.
(b) increases with increase in and .
(c) is such that its magnitude increases as goes from to . In fact, when and are parallel, the force on the particle is zero. For fixed magnitudes of and , the force is the maximum when .
(d) increases with increase in charge.
This suggests the dependence
which has been confirmed experimentally. In fact, the relation is (exactly),
The direction of is found out to satisfy the right hand thumb rule. Holding out your thumb use your right hand fingers to map out the rotation from to . The direction of is given by the direction in which the thumb points.
Now, since is a vector with direction perpendicular to both and , we write the expression for as
where the vector , the cross product of and , is understood to be a vector such that its magnitude is and its direction is given by the right hand thumb rule
In general, the cross product of and , i.e. is a vector with magnitude ( being the angle between and ) and direction perpendicular to the plane of and such that , and this direction form a right handed system.
It is important to keep in mind that the cross product is a vector; the dot product was a scalar. The cross product is also referred to as the vector product.
The cross product of and , say , has an interesting geometrical interpretation. Since represents the area of the parallelogram with adjacent sides and :
In fact, the area of the parallelogram can itself be treated as a vector (as it is in physical phenomena):
The area of the triangle formed with and as two sides is simply .