ONE MORE STEP TOWARD BETTER TOMORROW : CHAPTER 20 -Cross Product

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 $\vec v$ 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 $\theta = \dfrac{\pi }{2}$ .

(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 $\dfrac{1}{2}\left| {\vec A} \right| = \dfrac{1}{2}\;\left| {\vec a \times \vec b} \right|$ .

Friday, 8 August 2014

CHAPTER 20 -Cross Product

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