Question

If $a$ is perpendicular to $b$ and $c$, $\left|a\right|=2$, $\left|b\right|=3$, $\left|c\right|=4$ and the angle between $b$ and $c$ is $\frac{2\pi }{3}$ , then $[abc]$ is equal to?

• A. $4\surd 3$
• B. $6\surd 3$
• C. $12\surd 3$
• D. $18\surd 3$

Answer 1

The correct option is C

$12\surd 3$

Explanation for the correct option:

Find the value of $[abc]$:

Given, $\left|a\right|=2$, $\left|b\right|=3$, $\left|c\right|=4$

$\because a\perp b$ and $a\perp c$

$\begin{array}{rcl}\therefore |abc|& =& a·(b\times c)\\ & =& \left|a\right||b\times c|\cos 0\\ & =& \left|a\right||b\times c|\\ & =& \left|a\right|\left|b\right|\left|c\right|\sin \frac{2\pi }{3}\\ & =& 2\times 3\times 4\times \frac{\sqrt{3}}{2}\\ & =& \mathbf{12}\sqrt{\mathbf{3}}\end{array}$ $$\begin{gathered} \begin{array}{rcl}& & \left[\mathbf{\because }\mathbf{|}\mathbf{b}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{c}\mathbf{|}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathbf{b}\mathbf{\right|}\mathbf{}\mathbf{\left|}\mathbf{c}\mathbf{\right|}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }\mathbf{} \\ \mathit{a}\mathbf{·}\mathit{b}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\left|}\mathbf{a}\mathbf{\right|}\mathbf{}\mathbf{\left|}\mathbf{b}\mathbf{\right|}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }\right]\end{array} \end{gathered}$$

Hence, Option ‘C’ is Correct.