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$\vec{A}$ and $\overline{\mathrm{B}}$ are two vectors and $\theta$ is the angle between them, if $|\overrightarrow{\mathrm{A}} \times \overline{\mathrm{B}}|=$ $\sqrt{3}(\overrightarrow{\mathrm{A}} \cdot \overline{\mathrm{B}})$, the value of $\theta$ is.
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Verified Answer
The correct answer is:
$60^{\circ}$
Finding the angle between the vectors:
\(\begin{aligned}
& |\mathrm{A} \times \mathrm{B}|=\sqrt{3}(\mathrm{AB}) \\
& |\mathrm{A}||\mathrm{B}| \sin (\theta)=\sqrt{3}|\mathrm{~A}||\mathrm{B}| \cos (\theta)
\end{aligned}\)
Cancelling both sides we get
\(\begin{aligned}
& \sin (\theta)=\sqrt{3} \cos (\theta) \\
& \frac{\sin (\theta)}{\cos (\theta)}=\sqrt{3} \\
& \tan (\theta)=\sqrt{3}
\end{aligned}\)
Hence \(\theta=60^{\circ}\)
Hence the angle between A and B is \(60^{\circ}\)
\(\begin{aligned}
& |\mathrm{A} \times \mathrm{B}|=\sqrt{3}(\mathrm{AB}) \\
& |\mathrm{A}||\mathrm{B}| \sin (\theta)=\sqrt{3}|\mathrm{~A}||\mathrm{B}| \cos (\theta)
\end{aligned}\)
Cancelling both sides we get
\(\begin{aligned}
& \sin (\theta)=\sqrt{3} \cos (\theta) \\
& \frac{\sin (\theta)}{\cos (\theta)}=\sqrt{3} \\
& \tan (\theta)=\sqrt{3}
\end{aligned}\)
Hence \(\theta=60^{\circ}\)
Hence the angle between A and B is \(60^{\circ}\)
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