Question

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0},\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=5,\left|\overrightarrow{c}\right|=7,$ then the angle between $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$is
(a) $\frac{\mathrm{\pi }}{6}$

(b) $\frac{2\mathrm{\pi }}{3}$

(c) $\frac{5\mathrm{\pi }}{3}$

(d) $\frac{\mathrm{\pi }}{3}$

Answer 1

(d) $\frac{\mathrm{\pi }}{3}$

$$\begin{gathered} \text{Given},\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=5\mathrm{and}\left|\overrightarrow{c}\right|=7...\left(\mathrm{i}\right) \\ \text{Let θ be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}\text{.} \\ \text{Given that} \\ \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0 \\ \Rightarrow \overrightarrow{a}+\overrightarrow{b}=-\overrightarrow{c} \\ \Rightarrow \left|\overrightarrow{a}+\overrightarrow{b}\right|={\left|-\overrightarrow{c}\right|}^2 \\ \Rightarrow {\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+2\overrightarrow{a}.\overrightarrow{b}={\left|\overrightarrow{c}\right|}^2 \\ \Rightarrow 2\overrightarrow{a}.\overrightarrow{b}={\left|\overrightarrow{c}\right|}^2-{\left|\overrightarrow{a}\right|}^2-{\left|\overrightarrow{b}\right|}^2 \\ \Rightarrow 2\overrightarrow{a}.\overrightarrow{b}=7^2-3^2-5^2\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow 2\overrightarrow{a}.\overrightarrow{b}=15 \\ \Rightarrow 2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =15 \\ \Rightarrow 2\left(3\right)\left(5\right)\cos \theta =15\left[\mathrm{Using}\left(\mathrm{i}\right)\right] \\ \Rightarrow \cos \theta =\frac{1}{2} \\ \therefore \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$