Question

a = $\frac{i-2i}{\sqrt{5}}$ . $\overline{b}=\frac{2i+j+3\hat{k}}{\sqrt{14}}$then find $(2\overline{a}+\overline{b})$ $\{(\overline{a}\times \overline{b})\times (\overline{a}-2\overline{b})\}$ ?

Answer 1

Here $\overline{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}\overline{b}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}$

$Hereweneedtofind$

$\left(2\overline{a}+\overline{b}\right)\cdot \left\{\left(\overline{a}\times \overline{b}\right)\left(\overline{a}-2\overline{b}\right)\right\}=\left(\overline{a}-\overline{b}\right)\cdot \left\{\left(\overline{a}-2\overline{b}\right)\left(2\overline{a}\times \overline{b}\right)\right\}$

Since $\left(\overline{A}\cdot \left(\overline{B}\times \overline{C}\right)=\overline{B}\cdot \left(\overline{C}\times \overline{A}\right)\right)$

$=\left(\overline{a}\times \overline{b}\right)\cdot \left\{\overline{a}\times \overline{b}-4\overline{b}\times \overline{a}\right\}$

$=\left(\overline{a}\times \overline{b}\right)\cdot \left\{5\overline{a}\times \overline{b}\right\}$

$=5\cdot {\left|\overline{a}\times \overline{b}\right|}^2$

Here

$\overline{a}\times \overline{b}=\left|\begin{array}{c}\hat{i}\hat{j}\hat{k}\\ \frac{1}{5}\frac{-2}{5}0\\ \frac{2}{\sqrt{14}}\frac{1}{\sqrt{14}}\frac{3}{\sqrt{14}}\end{array}\right|$

$=\hat{i}\left(\frac{-6}{\sqrt{70}}\right)-\hat{j}\left(\frac{3}{\sqrt{70}}\right)+\hat{k}\left(\frac{5}{\sqrt{70}}\right)$

$=\frac{-6}{\sqrt{70}}\hat{i}-\frac{3}{\sqrt{70}}\hat{j}+\frac{5}{\sqrt{70}}\hat{k}$

$\left|\overline{a}\times \overline{b}\right|=\sqrt{\frac{36}{70}+\frac{9}{70}+\frac{25}{70}}$

where $\left|\overline{a}\times \overline{b}\right|=1$

$Here5\cdot {\left|\overline{a}\times \overline{b}\right|}^2$

$=5\cdot 1$

$Answeris5$