Question

If $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$are two vectors of the same magnitude inclined at an angle of 30°, such that $\overrightarrow{a}·\overrightarrow{b}=3,\mathrm{find}\left|\overrightarrow{a}\right|,\left|\overrightarrow{b}\right|.$

Answer 1

$$\begin{gathered} \text{Given that the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}{\text{is 30}}^0\text{.} \\ \text{Also,} \\ \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|\mathrm{and}\overrightarrow{a}.\overrightarrow{b}=3 \\ \text{We know that} \\ \overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \\ \Rightarrow 3=\left|\overrightarrow{a}\right|\left|\overrightarrow{a}\right|\cos 30 \\ \Rightarrow 3={\left|\overrightarrow{a}\right|}^2\left(\frac{\sqrt{3}}{2}\right) \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2=\frac{6}{\sqrt{3}}=2\sqrt{3} \\ \Rightarrow \left|\overrightarrow{a}\right|=\sqrt{2\sqrt{3}}=\left|\overrightarrow{b}\right| \\ \therefore \left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\sqrt{2\sqrt{3}} \end{gathered}$$