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If $\hat{a}, \hat{b}$ and $\hat{c}$ are unit vectors satisfying $\hat{a}-\sqrt{3} \hat{b}+\hat{c}=\overrightarrow{0}$, then the angle between the vectors $\hat{a}$ and $\hat{c}$ is :
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Verified Answer
The correct answer is:
$\frac{\pi}{3}$
$\frac{\pi}{3}$
Let angle between $\hat{a}$ and $\hat{c}$ be $\theta$.
Now, $\hat{a}-\sqrt{3} \hat{b}+\hat{c}=\overrightarrow{0}$
$$
\begin{aligned}
& \Rightarrow(\hat{a}+\hat{c})=\sqrt{3} \hat{b} \\
& \Rightarrow(\hat{a}+\hat{c}) \cdot(\hat{a}+\hat{c})=3(\hat{b} \cdot \hat{b}) \\
& \Rightarrow \hat{a} \cdot \hat{a}+\hat{a} \cdot \hat{c}+\hat{c} \cdot \hat{a}+\hat{c} \cdot \hat{c}=3 \times 1 \\
& \Rightarrow 1+2 \cos \theta+1=3 \\
& \Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}
\end{aligned}
$$
Now, $\hat{a}-\sqrt{3} \hat{b}+\hat{c}=\overrightarrow{0}$
$$
\begin{aligned}
& \Rightarrow(\hat{a}+\hat{c})=\sqrt{3} \hat{b} \\
& \Rightarrow(\hat{a}+\hat{c}) \cdot(\hat{a}+\hat{c})=3(\hat{b} \cdot \hat{b}) \\
& \Rightarrow \hat{a} \cdot \hat{a}+\hat{a} \cdot \hat{c}+\hat{c} \cdot \hat{a}+\hat{c} \cdot \hat{c}=3 \times 1 \\
& \Rightarrow 1+2 \cos \theta+1=3 \\
& \Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}
\end{aligned}
$$
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