Question

If $\left|\overrightarrow{A}\right|=2$ and $\left|\overrightarrow{B}\right|=5$ and $\left|\overrightarrow{A}·\overrightarrow{B}\right|=8$, then what is $\overrightarrow{A}·\overrightarrow{B}$ equal to?

• A. $6$
• B. $2$
• C. $20$
• D. $8$

Answer 1

The correct option is A

$6$

Expression for correct option:

Finding the value of $\overrightarrow{A}·\overrightarrow{B}$:

Given,

$\left|\overrightarrow{A}\right|=2$ and $\left|\overrightarrow{B}\right|=5$

$\left|\overrightarrow{A}·\overrightarrow{B}\right|=8$

We know that,

$|A\times B|=\left|A\right|\left|B\right|\sin \theta$

Therefore,

$$\begin{gathered} \Rightarrow 2\times 5\times \sin \theta =8 \\ \Rightarrow 10\times \sin \theta =8 \\ \Rightarrow \sin \theta =\frac{8}{10} \\ =\frac{4}{5} \end{gathered}$$

Also,

$\begin{array}{rcl}{\sin }^2\theta +{\cos }^2\theta & =& 1\\ & \Rightarrow & {\left(\frac{4}{5}\right)}^2+{\cos }^2\theta =1\\ & \Rightarrow & {\cos }^2\theta =1-{\left(\frac{4}{5}\right)}^2\\ & =& 1-\left(\frac{16}{25}\right)\\ & =& \left(\frac{25-16}{25}\right)\\ & =& \left(\frac{9}{25}\right)\\ & =& \sqrt{\frac{9}{25}}\\ & =& \frac{3}{5}\end{array}$

Now,

$\begin{array}{rcl}\overrightarrow{A}\cdot \overrightarrow{B}& =& \left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\cos \theta \\ & =& 2\times 5\times \frac{3}{5}\\ & =& 6\end{array}$

Hence, the correct option is C.