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If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are wit vectors such that $\mathbf{a}+\mathbf{b}+\mathbf{c}=0$ and $(\mathbf{a}, \mathbf{b})=\frac{\pi}{3}$, then $|\mathbf{a} \times \mathbf{b}|+|\mathbf{b} \times \mathbf{c}|+|\mathbf{c} \times \mathbf{a}|=$
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Verified Answer
The correct answer is:
$\frac{3 \sqrt{3}}{2}$
We have,
a, b, care unit vector
$\therefore|\mathbf{a}|=|\mathbf{b}|=|\mathbf{c}|=1$ and $\mathbf{a}+\mathbf{b}+\mathbf{c}=0$, angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\pi}{3}$.
Now,
$\mathbf{a}+\mathbf{b}+\mathbf{c}=0$
$\begin{aligned}
& \Rightarrow \quad \mathbf{a} \times\langle\mathbf{a}+\mathbf{b}+\mathbf{c}\rangle=0 \\
& \Rightarrow \mathbf{a} \times \mathbf{a}+\mathbf{a} \times \mathbf{b}+\mathbf{a} \times \mathbf{c}=0 \\
& \Rightarrow \quad \mathbf{a} \times \mathbf{b}=\mathbf{c} \times \mathbf{a} \\
& \Rightarrow \quad|\mathbf{a} \times \mathbf{b}|=|\mathbf{c} \times \mathbf{a}| \\
& \text {Similarly }|\mathbf{a} \times \mathbf{b}|=|\mathbf{b} \times \mathbf{c}| \\
& \therefore|\mathbf{a} \times \mathbf{b}|+|\mathbf{b} \times \mathbf{c}|+|\mathbf{c} \times \mathbf{a}| \\
& =3|\mathbf{a} \times \mathbf{b}| \\
& =3|\mathbf{a}||\mathbf{b}| \sin (a, b) \\
& =3 \times 1 \times 1 \times \sin \frac{\pi}{3}=\frac{3 \sqrt{3}}{2}
\end{aligned}$
a, b, care unit vector
$\therefore|\mathbf{a}|=|\mathbf{b}|=|\mathbf{c}|=1$ and $\mathbf{a}+\mathbf{b}+\mathbf{c}=0$, angle between $\mathbf{a}$ and $\mathbf{b}$ is $\frac{\pi}{3}$.
Now,
$\mathbf{a}+\mathbf{b}+\mathbf{c}=0$
$\begin{aligned}
& \Rightarrow \quad \mathbf{a} \times\langle\mathbf{a}+\mathbf{b}+\mathbf{c}\rangle=0 \\
& \Rightarrow \mathbf{a} \times \mathbf{a}+\mathbf{a} \times \mathbf{b}+\mathbf{a} \times \mathbf{c}=0 \\
& \Rightarrow \quad \mathbf{a} \times \mathbf{b}=\mathbf{c} \times \mathbf{a} \\
& \Rightarrow \quad|\mathbf{a} \times \mathbf{b}|=|\mathbf{c} \times \mathbf{a}| \\
& \text {Similarly }|\mathbf{a} \times \mathbf{b}|=|\mathbf{b} \times \mathbf{c}| \\
& \therefore|\mathbf{a} \times \mathbf{b}|+|\mathbf{b} \times \mathbf{c}|+|\mathbf{c} \times \mathbf{a}| \\
& =3|\mathbf{a} \times \mathbf{b}| \\
& =3|\mathbf{a}||\mathbf{b}| \sin (a, b) \\
& =3 \times 1 \times 1 \times \sin \frac{\pi}{3}=\frac{3 \sqrt{3}}{2}
\end{aligned}$
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