Question

If $|\overrightarrow{a}|=2,|\overrightarrow{b}|=5$ and $|\overrightarrow{a}\times \overrightarrow{b}|=8$, then what is $\overrightarrow{a}\cdot \overrightarrow{b}$ equal to?

• A. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

The correct option is A $6$

Given

$\left|\overrightarrow{a}\right|=2$

$\left|\overrightarrow{b}\right|=5$

$\Rightarrow |\overrightarrow{a}\times \overrightarrow{b}|=8$

$\Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta =8$

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

$\Rightarrow 10\sin \theta =8$

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

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

$\Rightarrow \cos \theta =\sqrt{1-(\frac{4}{5}{)}^2}$

$\Rightarrow \cos \theta =\sqrt{1-\frac{16}{25}}$

$\Rightarrow \cos \theta =\sqrt{\frac{25-16}{25}}$

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

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

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

$\Rightarrow \Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=2\times 5\times \frac{3}{5}$

$\Rightarrow \overrightarrow{a}\cdot \overrightarrow{b}=6$

Hence option A is correct