The dot product discussed in the previous section, was introduced through the requirement that arose in calculating the work done by a given force  \(\vec F\) when the point of application of the force is displaced by a certain amount given by \(\vec s\) :

\[W = \vec F \cdot \vec s\]

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 \(\vec a\,\,{\text{and}}\,\,\vec b\)  is another vector \(\vec c\)  and the relation is represented as

\[\vec c = \vec a \times \vec b\]

Let us, through a physical example, understand what the cross product means.

Consider a horizontal magnetic field, which we can represent by \(\vec B\)  , and a charge q projected into this field with a velocity \(\vec v\)(at an angle  \(\theta \) with the horizontal).

Experiments show that the force \(\vec F\) acting on this particle

(a) is perpendicular to the plane of \(\vec v\) and \(\vec B\)  and goes into the plane for the figure above.

(b) increases with increase in \(\left| {\vec v} \right|\;\;{\text{and}}\;\;\left| {\vec B} \right|\).

(c) is such that its magnitude increases as \(\theta \) goes from \(0\;\;{\text{to}}\;\frac{\pi }{2}\). In fact, when \(\vec v\) and \(\vec B\) are parallel, the force on the particle is zero. For fixed magnitudes of \(\vec v\) and \(\vec B\), the force is the maximum when \(\begin{align}\theta  = \frac{\pi }{2}\end{align}\) .

(d) increases with increase in charge.

This suggests the dependence

\[\left| {\vec F} \right|\; \propto q\;\left| {\vec v} \right|\;\left| {\vec B} \right|\;\sin \theta \]

which has been confirmed experimentally. In fact, the relation is (exactly),

\[\left| {\vec F} \right| = q\;\left| {\vec v} \right|\;\left| {\vec B} \right|\;\sin \theta \]

The direction of \(\vec F\) 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 \(\vec v\) to \(\vec B\). The direction of  \(\vec F\) is given by the direction in which the thumb points.

Now, since \(\vec F\)  is a vector with direction perpendicular to both \(\vec v\) and \(\vec B\), we write the expression for \(\vec F\) as 

\[\boxed{\vec F = q(\vec v \times \vec B)}\]

where the vector  \(\vec v \times \vec B\), the cross product of \(\vec v\) and \(\vec B\) , is understood to be a vector such that its magnitude is \(\left| {\vec v} \right|\left| {\vec B} \right|\sin \theta .\) and its direction is given by the right hand thumb rule

In general, the cross product of \(\vec a\,\,and\,\,\vec b\) , i.e.\(\vec c = \vec a \times \vec b\) is a vector with magnitude \(\left| {\vec a} \right|\;\left| {\vec b} \right|\sin {{\theta }}\)( \({{\theta }}\) being the angle between \(\vec a\,\,and\,\,\vec b\) ) and direction perpendicular to the plane of \(\vec a\,\,and\,\,\vec b\) such that \(\vec a\,\,,\,\,\vec b\) 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  \(\vec a\,\,{\text{and}}\,\,\vec b\) , say \(\vec c\), has an interesting geometrical interpretation. Since \(\left| {\vec c} \right| = \left| {\vec a} \right|\;\left| {\vec b} \right|\;\sin \theta ,\,\,\left| {\vec c} \right|\) represents the area of the parallelogram with adjacent sides \(\vec a\,\,{\text{and}}\,\,\vec b\) :

In fact, the area of the parallelogram can itself be treated as a vector (as it is in physical phenomena):

\[\vec A = \vec a \times \vec b\]

The area of the triangle formed with \(\vec a\,\,{\text{and}}\,\,\vec b\) as two sides is simply \(\begin{align}\frac{1}{2}\left| {\vec A} \right| = \frac{1}{2}\;\left| {\vec a \times \vec b} \right|\end{align}\) .

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