Question

The angle between two vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$ with magnitudes $\sqrt{3}$ and 4 respectively and $\overrightarrow{a}·\overrightarrow{b}=2\sqrt{3},\mathrm{is}$
$$\begin{gathered} \left(\mathrm{a}\right)\frac{\mathrm{\pi }}{6} \\ \left(\mathrm{b}\right)\frac{\mathrm{\pi }}{3} \\ \left(\mathrm{c}\right)\frac{\mathrm{\pi }}{2} \\ \left(\mathrm{d}\right)\frac{5\mathrm{\pi }}{2} \end{gathered}$$

Answer 1

Given:
$$\begin{gathered} \left|\overrightarrow{a}\right|=\sqrt{3} \\ \left|\overrightarrow{b}\right|=4 \\ \overrightarrow{a}·\overrightarrow{b}=2\sqrt{3} \end{gathered}$$


$$\begin{gathered} \overrightarrow{a}·\overrightarrow{b}=2\sqrt{3} \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =2\sqrt{3} \\ \Rightarrow \sqrt{3}\times 4\times \cos \theta =2\sqrt{3} \\ \Rightarrow \cos \theta =\frac{2\sqrt{3}}{4\sqrt{3}} \\ \Rightarrow \cos \theta =\frac{1}{2} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$

Hence, the correct option is (b).