Question

If $|\overrightarrow{a}|=2,|\overrightarrow{b}|=7$ and $\overline{a}\times \overrightarrow{b}=3\overline{i}+2\overline{j}+6\overline{k}$, then $\overline{a}\cdot \overline{b}=?$

• A. $\sqrt{147}$
• B. $\sqrt{145}$
• C. $\sqrt{143}$
• D. $\sqrt{142}$

Answer 1

The correct option is A $\sqrt{147}$

$|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{(3{)}^2+(2{)}^2+(6{)}^2}=7$

$|\overrightarrow{a}\times \overrightarrow{b}|=|\overrightarrow{a}||\overrightarrow{b}|\sin \theta$

where $\theta$is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$

$7=2\times 7\sin \theta$

$\sin \theta =\frac{1}{2}$

$\theta ={30}^{\circ }$

$\overrightarrow{a}\cdot \overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \theta$

$=2\times 7\times \cos {30}^{\circ }$

$=14\times \frac{\sqrt{3}}{2}$

$=\sqrt{147}$