Question

$\overline{a}$ is perpendicular to both $\overline{b}$ and $\overline{c}$. The angle between $\overline{b}$ and $\overline{c}$ is $\frac{2\pi }{3}$. If $\left|\overline{a}\right|=2,\left|\overline{b}\right|=3,\left|\overline{c}\right|=4$, then $\overline{c}.(\overline{a}\times \overline{b})=$

• A. $18\sqrt{3}$
• B. $12\sqrt{3}$
• C. $8\sqrt{3}$
• D. $6\sqrt{3}$

Answer 1

The correct option is B $12\sqrt{3}$

$\overline{c}.(\overline{a}\times \overline{b})$ can be expressed as:

${\left[\overline{c}\overline{a}\overline{b}\right]}^2=\left|\begin{array}{ccc}\overline{a}.\overline{a}& \overline{a}.\overline{b}& \overline{a}.\overline{c}\\ \overline{b}.\overline{a}& \overline{b}.\overline{b}& \overline{b}.\overline{c}\\ \overline{c}.\overline{a}& \overline{c}.\overline{b}& \overline{c}.\overline{c}\end{array}\right|=\left|\begin{array}{ccc}2\times 2& 0& 0\\ 0& 3\times 3& 3\times 4\times \cos \left({120}^{\circ }\right)\\ 0& 3\times 4\times \cos \left({120}^{\circ }\right)& 4\times 4\end{array}\right|=432\left[\overline{c}\overline{a}\overline{b}\right]=\sqrt{4\times 108}=12\sqrt{3}$