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Friday, 8 August 2014

CHAPTER 20 -Cross Product

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 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| and \;\left| {\vec B} \right|.
(c) is such that its magnitude increases as \theta  goes from 0 to \dfrac{\pi }{2} . In fact, when \vec vand \vec B are parallel, the force on the particle is zero. For fixed magnitudes of \vec vand \vec B, the force is the maximum when \theta  = \dfrac{\pi }{2}.
(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
{\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 {\rm{\theta }} ( {\rm{\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 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 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 and \vec b as two sides is simply \dfrac{1}{2}\left| {\vec A} \right| = \dfrac{1}{2}\;\left| {\vec a \times \vec b} \right|.
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