Question

Find $\overrightarrow{A}\cdot \overrightarrow{B}$ if $|\overrightarrow{A}|=2,|\overrightarrow{B}|=5$ and $|\overrightarrow{A}\times \overrightarrow{B}|=8$.

Answer 1

Given: $|\overrightarrow{A}|=2,|\overrightarrow{B}|=5$

$\Rightarrow \overrightarrow{A}\times \overrightarrow{B}=|A||B|\sin \theta$

$5\times 2\sin \theta =8$

$\Rightarrow \sin \theta =\frac{4}{5}$

But $\cos \theta =\sqrt{1-{\sin }^2\theta }$

$\Rightarrow \cos \theta =\sqrt{1-\frac{116}{25}}=\sqrt{\frac{9}{25}}$

$\Rightarrow \cos \theta =\pm \frac{3}{5}$

$\overrightarrow{A}\cdot \overrightarrow{B}=|A||B|\cos \theta =\pm 2\times 5\times \frac{3}{5}=\pm 6$

Thus, $\overrightarrow{A}\cdot \overrightarrow{B}=\pm 6$