Q91 / 160 · TS EAMCET 2017 · Mathematics
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 $|\mathbf{a} \times \mathbf{b}|+|\mathbf{b} \times \mathbf{c}|+|\mathbf{c} \times \mathbf{a}|=$
- A$\frac{3}{2}$
- B0
- Correct answerC$\frac{3 \sqrt{3}}{2}$
- D3
Solution
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,
$\mathbf{a}+\mathbf{b}+\mathbf{c}=0$
$\begin{aligned}<br />& \Rightarrow \quad \mathbf{a} \times\langle\mathbf{a}+\mathbf{b}+\mathbf{c}\rangle=0 \\<br />& \Rightarrow \mathbf{a} \times \mathbf{a}+\mathbf{a} \times \mathbf{b}+\mathbf{a} \times \mathbf{c}=0 \\<br />& \Rightarrow \quad \mathbf{a} \times \mathbf{b}=\mathbf{c} \times \mathbf{a} \\<br />& \Rightarrow \quad|\mathbf{a} \times \mathbf{b}|=|\mathbf{c} \times \mathbf{a}| \\<br />& \text {Similarly }|\mathbf{a} \times \mathbf{b}|=|\mathbf{b} \times \mathbf{c}| \\<br />& \therefore|\mathbf{a} \times \mathbf{b}|+|\mathbf{b} \times \mathbf{c}|+|\mathbf{c} \times \mathbf{a}| \\<br />& =3|\mathbf{a} \times \mathbf{b}| \\<br />& =3|\mathbf{a}||\mathbf{b}| \sin (a, b) \\<br />& =3 \times 1 \times 1 \times \sin \frac{\pi}{3}=\frac{3 \sqrt{3}}{2}<br />\end{aligned}$
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,
$\mathbf{a}+\mathbf{b}+\mathbf{c}=0$
$\begin{aligned}<br />& \Rightarrow \quad \mathbf{a} \times\langle\mathbf{a}+\mathbf{b}+\mathbf{c}\rangle=0 \\<br />& \Rightarrow \mathbf{a} \times \mathbf{a}+\mathbf{a} \times \mathbf{b}+\mathbf{a} \times \mathbf{c}=0 \\<br />& \Rightarrow \quad \mathbf{a} \times \mathbf{b}=\mathbf{c} \times \mathbf{a} \\<br />& \Rightarrow \quad|\mathbf{a} \times \mathbf{b}|=|\mathbf{c} \times \mathbf{a}| \\<br />& \text {Similarly }|\mathbf{a} \times \mathbf{b}|=|\mathbf{b} \times \mathbf{c}| \\<br />& \therefore|\mathbf{a} \times \mathbf{b}|+|\mathbf{b} \times \mathbf{c}|+|\mathbf{c} \times \mathbf{a}| \\<br />& =3|\mathbf{a} \times \mathbf{b}| \\<br />& =3|\mathbf{a}||\mathbf{b}| \sin (a, b) \\<br />& =3 \times 1 \times 1 \times \sin \frac{\pi}{3}=\frac{3 \sqrt{3}}{2}<br />\end{aligned}$
Hence, the correct answer is option: 'C'.