Question

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are vector is space given by $\overrightarrow{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\overrightarrow{b}=\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}},$ then the value of $(2\overrightarrow{a}+\overrightarrow{b}).[(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{a}-2\overrightarrow{b})]$ is

• A. $5$
• B. $3$
• C. $-5$
• D. $-3$

Answer 1

The correct option is A $5$

Given: $\overrightarrow{a}=\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$

$\left|\overrightarrow{a}\right|=\left|\frac{\hat{i}-2\hat{j}}{\sqrt{5}}\right|=\frac{\sqrt{1+4}}{\sqrt{5}}=\frac{\sqrt{5}}{\sqrt{5}}=1$

$\left|\overrightarrow{b}\right|=\left|\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}\right|=\left|\frac{\sqrt{4+1+9}}{\sqrt{14}}\right|=\frac{\sqrt{14}}{\sqrt{14}}=1$

$\overrightarrow{a}.\overrightarrow{b}=\left(\frac{\hat{i}-2\hat{j}}{\sqrt{5}}\right)\left(\frac{2\hat{i}+\hat{j}+3\hat{k}}{\sqrt{14}}\right)=2-2+0=0$

$(2\overrightarrow{a}+\overrightarrow{b})[(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{a}-2\overrightarrow{b})]$

$=(2\overrightarrow{a}+\overrightarrow{b})[\overrightarrow{b}+2\overrightarrow{a}]$

$={\left|\overrightarrow{b}\right|}^2+4{\left|\overrightarrow{a}\right|}^2=4+1=5$