Vector (or Cross) Product of Two Vectors

VECTOR (OR CROSS) PRODUCT OF TWO VECTORS

The vector product of two vectors $\bar{a}$and $\bar{b}$, denoted by $\bar{a}\times \bar{b}$, is defined as the vector $|\bar{a}||\bar{b}|\sin \theta \hat{n}$, where $\theta$ is the angle between the vectors $\bar{a}$

and $\bar{b}$and $\hat{n}$ is a unit vector perpendicular to both $\bar{a}$and $\bar{b}$(i.e., perpendicular to the plane of $\bar{a}$and $\bar{b}$).The sense of $\hat{n}$ is obtained by the right hand thumb rule i.e.$\bar{a}$, $\bar{b}$ and $\hat{n}$ form a right-handed screw.

If we curl the fingers of our right hand from $\bar{a}$to $\bar{b}$through the smaller angle (keeping the initial point of $\bar{a}$ and $\bar{b}$same), the thumb points in the direction of $\hat{n}$. In this case, $\bar{a}$,$\bar{b}$and $\hat{n}$ (or $\bar{a}\times \bar{b}$), in that order are said to form a right handed system. It is evident that $|\bar{a}\times \bar{b}|$= absin$\theta$.

Properties:

$\bullet$ $\bar{a}\times \bar{a}=\bar{o}$$\Rightarrow$ $\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times \hat{k}=\bar{o}$$\bullet$ $\bar{a}\times \bar{b}=-(\bar{b}\times \bar{a})$ (non-commutative)$\bullet$ $\bar{a}\times (\bar{b}+\bar{c})=\bar{a}\times \bar{b}+\bar{a}\times \bar{c}$ (Distributive)$\bullet$ $(l\bar{a})\times (m\bar{b})=lm(\bar{a}\times \bar{b})$$\bullet$ $\bar{a}\times \bar{b}=\bar{o}$ $\iff$ $\bar{a}and\bar{b}$ are collinear (if none of $\bar{a},or\bar{b}$is a zero vector)$\bullet$ $\hat{i}\times \hat{j}=\hat{k},\hat{j}\times \hat{k}=\hat{i},\hat{k}\times \hat{i}=\hat{j}$;;;;;$\hat{j}\times \hat{i}=-\hat{k},\hat{k}\times \hat{j}=-i,\hat{i}\times \hat{k}=-\hat{j}$$\bullet$ If $a=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}and{b}_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ then \overline a \, \times \overline b;= \,\left| {\begin{array}{*{20}{c}};{\hat i}{\hat j}{\hat k} \\;;{{a_1}}{{a_2}}{{a_3}} \\;;{{b_1}}{{b_2}}{{b_3}};\end{array}} \right|;;;;;;;;;;;= ;;;$\left(a_1{b}_3-a_3+b_2\right)\hat{i}+\left(a_3{b}_1+a_1{b}_3\right)\hat{j}\left(a_1{b}_2-a_2{b}_1\right)\hat{K}$;;;;;$\bullet$Any vector perpendicular to the plane of $\overline{a}and\overline{d}$ is $\lambda$ ($\bar{a}\times \bar{b}$) where $\lambda$ is a real number. Unit vector perpendicular to $\bar{a}and\bar{b}$ is $\pm$ $\frac{\bar{a}\times \bar{b}}{|\bar{a}\times \bar{b}|}$ $\bullet$ $|\bar{a}\times \bar{b}|$denotes the area of the parallelogram OACB, whereas area of $\Delta$OAB = $\frac{1}{2}|\bar{a}\times \bar{b}|$Area is also treated as a vector with its direction in the proper sense.

$\Rightarrow$ $|\vec{b}{|}^2+4|\vec{c}{|}^2-4\vec{b}\cdot \vec{c}={\lambda }^2|\vec{a}{|}^2$$\Rightarrow$16 + 4 – 4(4) (1) $\frac{1}{4}$ = $\lambda$²(1)

$\Rightarrow$ $\lambda$² = 16 $\Rightarrow$ $\lambda$ $\pm$ 4.