If $\bar{a}$ is perpendicular to $\bar{b}$ and $\bar{c},|\vec{a}|=2$, $|\bar{b}|=3,|\bar{c}|=4$ and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{3}$, then $\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]=$

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Aicharya · Accepted Answer

Let $\hat{\mathrm{n}}$ be the unit vector perpendicular to $\overline{\mathrm{b}}$ and $\bar{c}$. $\begin{aligned} {\left[\begin{array}{lll} \overline{\mathrm{a}} & \overline{\mathrm{~b}} & \overline{\mathrm{c}} \end{array}ight] } & =\overline{\mathrm{a}} \cdot(\overline{\mathrm{~b}} 	imes \overline{\mathrm{c}}) \ & =\overline{\mathrm{a}} \cdot(|\overline{\mathrm{~b}}||\bar{c}| \sin 	heta \hat{\mathrm{n}}) \ & =\overline{\mathrm{a}} \cdot\left(3 	imes 4 \sin \frac{\pi}{3} \cdot \hat{\mathrm{n}}ight) \ & =\overline{\mathrm{a}} \cdot\left(12 	imes \frac{\sqrt{3}}{2} \hat{\mathrm{n}}ight) \ & =6 \sqrt{3}|\overrightarrow{\mathrm{a}}| \hat{\mathrm{n}} \mid \cos 0 \ & =6 \sqrt{3} 	imes 2 	imes 1 \ & =12 \sqrt{3} \end{aligned}$

If $\bar{a}$ is perpendicular to $\bar{b}$ and $\bar{c},|\vec{a}|=2$, $|\bar{b}|=3,|\bar{c}|=4$ and the…

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