Question

Find the angle between two vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$if
(i) $\left|\overrightarrow{a}\right|=\sqrt{3},\left|\overrightarrow{b}\right|=2\mathrm{and}\overrightarrow{a}·\overrightarrow{b}=\sqrt{6}$

(ii) $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=3\mathrm{and}\overrightarrow{a}·\overrightarrow{b}=1$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\text{Let θ be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}\text{.} \\ \text{Given that} \\ \left|\overrightarrow{a}\right|=\sqrt{3},\left|\overrightarrow{b}\right|=2\mathrm{and}\overrightarrow{a}.\overrightarrow{b}=\sqrt{6}...\left(1\right) \\ \text{We know that} \\ \overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \\ \Rightarrow \sqrt{6}=\left(\sqrt{3}\right)\left(2\right)\cos \theta \left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow \cos \theta =\frac{\sqrt{6}}{2\sqrt{3}}=\frac{1}{\sqrt{2}} \\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\mathrm{\pi }}{4} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\text{Let θ be the angle between}\overrightarrow{a}\text{and}\overrightarrow{b}\text{.} \\ \text{Given that} \\ \left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=3\mathrm{and}\overrightarrow{a}.\overrightarrow{b}=1...\left(1\right) \\ \text{We know that} \\ \overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta \\ \Rightarrow 1=\left(3\right)\left(3\right)\cos \theta \left[\mathrm{Using}\left(1\right)\right] \\ \Rightarrow \cos \theta =\frac{1}{\left(3\right)\left(3\right)}=\frac{1}{9} \\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{1}{9}\right) \end{gathered}$$