Mathematics - Vector Question with Solution | TestHub

Given, $\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$

$\Rightarrow \overrightarrow{\mathrm{a}}\times \left(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\right)=\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}$

$\Rightarrow \left(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}\right)\overrightarrow{\mathrm{a}}-\left(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{a}}\right)\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}$

$\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 1& 1& 1\\ 0& 1& -1\end{array}\right|=-2\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\begin{array}{}\\ \\ \end{array}$

$\Rightarrow 3\overrightarrow{\mathrm{a}}-3\overrightarrow{\mathrm{b}}=-2\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$

$\Rightarrow 3\hat{\mathrm{i}}+3\hat{\mathrm{j}}+3\hat{\mathrm{k}}-3\overrightarrow{\mathrm{b}}=-2\mathrm{ }\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$

$\Rightarrow \overrightarrow{\mathrm{b}}=\frac{1}{3}\left(5\hat{\mathrm{i}}+2\hat{\mathrm{j}}+2\hat{\mathrm{k}}\right)$

$\therefore \left|\overrightarrow{\mathrm{b}}\right|=\frac{\sqrt{25+4+4}}{3}$

$\Rightarrow \left|\overrightarrow{\mathrm{b}}\right|=\sqrt{\frac{11}{3}}$