Vector Product of Two Vectors
Category : JEE Main & Advanced
(1) Definition :
The vector product or cross product of two vectors is defined as a vector
having a magnitude equal to the product of the magnitudes of two vectors with
the sine of angle between them, and direction perpendicular to the plane
containing the two vectors in accordance with right hand screw rule.
\[\overrightarrow{C}=\overrightarrow{A}\times \overrightarrow{B}\]
Thus,
if \[\overrightarrow{A}\] and \[\overrightarrow{B}\] are two vectors, then
their vector product written as \[\overrightarrow{A}\times \overrightarrow{B}\]
is a vector \[\overrightarrow{C}\] defined by
\[\overrightarrow{C}=\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta
\,\hat{n}\]
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The
direction of \[\overrightarrow{A}\times \overrightarrow{B},\] i.e.
\[\overrightarrow{C}\] is perpendicular to the plane containing vectors
\[\overrightarrow{A}\] and \[\overrightarrow{B}\] and in the sense of advance
of a right handed screw rotated from \[\overrightarrow{A}\] (first vector) to
\[\overrightarrow{B}\] (second vector) through the smaller angle between them.
Thus, if a right handed screw whose axis is perpendicular to the plane framed
by \[\overrightarrow{A}\] and \[\overrightarrow{B}\] is rotated from
\[\overrightarrow{A}\] to \[\overrightarrow{B}\] through the smaller angle
between them, then the direction of advancement of the screw gives the
direction of \[\overrightarrow{A}\times \overrightarrow{B}\] i.e.
\[\overrightarrow{C}\]
(2) Properties
(i)
Vector product of any two vectors is always a vector perpendicular to the plane
containing these two vectors, i.e., orthogonal to both the vectors
\[\overrightarrow{A}\] and \[\overrightarrow{B},\]though the vectors
\[\overrightarrow{A}\] and \[\overrightarrow{B}\] may or may not be orthogonal.
(ii)
Vector product of two vectors is not commutative, i.e.,
\[\overrightarrow{A}\times \overrightarrow{B}\ne \overrightarrow{B}\times
\overrightarrow{A}\] [but \[=-\overrightarrow{B}\times \overrightarrow{A}]\]
Here
it is worthy to note that
\[|\overrightarrow{A}\times
\overrightarrow{B}|=|\overrightarrow{B}\times \overrightarrow{A}|=AB\sin \theta
\]
i.e. in
case of vector \[\overrightarrow{A}\times \overrightarrow{B}\] and
\[\overrightarrow{B}\times \overrightarrow{A}\] magnitudes are equal but
directions are opposite.
(iii)
The vector product is distributive when the order of the vectors is strictly
maintained, i.e.
\[\overrightarrow{A}\times
(\overrightarrow{B}+\overrightarrow{C})=\overrightarrow{A}\times
\overrightarrow{B}+\overrightarrow{A}\times \overrightarrow{C}\]
(iv)
The vector product of two vectors will be maximum when \[\sin \theta =\max
=1,\] i.e., \[\theta ={{90}^{o}}\]
\[{{[\overrightarrow{A}\times
\overrightarrow{B}]}_{\max }}=AB\,\,\hat{n}\]
i.e.
vector product is maximum if the vectors are orthogonal.
(v) The vector product of two non- zero vectors will be
minimum when \[|\sin \theta |\,=\] minimum = 0, i.e., \[\theta
={{0}^{o}}\] or \[{{180}^{o}}\]
\[{{[\overrightarrow{A}\times \overrightarrow{B}]}_{\min
}}=0\]
i.e.
if the vector product of two non-zero vectors vanishes, the vectors are collinear.
(vi)
The self cross product, i.e., product of a vector by itself
vanishes, i.e., is null vector \[\overrightarrow{A}\times
\overrightarrow{A}=AA\sin {{0}^{o}}\,\hat{n}=\overrightarrow{0\,}\]
(vii)
In case of unit vector \[\hat{n}\times \hat{n}=\overrightarrow{0}\]so that
\[\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times
\hat{k}=\overrightarrow{0}\]
(viii)
In case of orthogonal unit vectors, \[\hat{i},\,\hat{j},\,\hat{k}\] in
accordance with right hand screw rule :
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\[\hat{i}\times
\hat{j}=\hat{k},\]
\[\hat{j}\times \hat{k}=\hat{i}\]
and \[\hat{k}\times
\hat{i}=\hat{j}\]
And as
cross product is not commutative,
\[\hat{j}\times \hat{i}=-\hat{k}\], \[\hat{k}\times \hat{j}=-\hat{i}\]and
\[\hat{i}\times \hat{k}=-\hat{j}\]
(x) In
terms of components
\[\overrightarrow{A}\times
\overrightarrow{B}=\left| \begin{matrix}
{\hat{i}} & {\hat{j}} & {\hat{k}} \\
{{A}_{x}} & {{A}_{y}} & {{A}_{z}} \\
{{B}_{x}} & {{B}_{y}} & {{B}_{z}} \\
\end{matrix}
\right|\]
\[=\hat{i}({{A}_{y}}{{B}_{z}}-{{A}_{z}}{{B}_{y}})\]\[+\hat{j}({{A}_{z}}{{B}_{x}}-{{A}_{x}}{{B}_{z}})+\hat{k}({{A}_{x}}{{B}_{y}}-{{A}_{y}}{{B}_{x}})\]
(3) Example
: Since vector product of two
vectors is a vector, vector physical quantities (particularly representing
rotational effects) like torque, angular momentum, velocity and force on a
moving charge in a magnetic field and can be expressed as the vector product of
two vectors. It is well ? established in physics that :
(i)
Torque \[\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}\]
(ii)
Angular momentum \[\overrightarrow{L}=\overrightarrow{r}\times
\overrightarrow{p}\]
(iii)
Velocity \[\overrightarrow{v}=\overrightarrow{\omega }\times
\overrightarrow{r}\]
(iv)
Force on a charged particle q moving with velocity \[\overrightarrow{v}\] in a
magnetic field \[\overrightarrow{B}\] is given by
\[\overrightarrow{F}=q(\overrightarrow{v}\times \overrightarrow{B})\]
(v)
Torque on a dipole in a field \[\overrightarrow{{{\tau
}_{E}}}=\overrightarrow{p}\times \overrightarrow{E}\]
and\[\overrightarrow{{{\tau }_{B}}}=\overrightarrow{M}\times
\overrightarrow{B}\]
