Mathematics - Vector Question with Solution | TestHub

Let angle between $\hat{a}$ and $\hat{c}$ be $\theta$. Now, $\hat{a}-\sqrt{3}\hat{b}+\hat{c}=\overrightarrow{0}$ $$\begin{array}{rl}& \Rightarrow (\hat{a}+\hat{c})=\sqrt{3}\hat{b}\\ & \Rightarrow (\hat{a}+\hat{c})\cdot (\hat{a}+\hat{c})=3(\hat{b}\cdot \hat{b})\\ & \Rightarrow \hat{a}\cdot \hat{a}+\hat{a}\cdot \hat{c}+\hat{c}\cdot \hat{a}+\hat{c}\cdot \hat{c}=3\times 1\\ & \Rightarrow 1+2\cos \theta +1=3\\ & \Rightarrow \cos \theta =\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}\end{array}$$