Question

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

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

Answer 1

Answer: (B)

Given,
$a\cdot b=0$
$a\cdot c=0$
$\therefore a\Vert b\times c$
and $|b\times c|=|b|\cdot |c|\sin \left(\frac{2\pi }{3}\right)$
$[\because |a|=2,|b|=3$ and $|c|=4]$
$=3\times 4\times \frac{\sqrt{3}}{2}=6\sqrt{3}$
Hence,$c\cdot (a\times b)=a\cdot (b\times c)[\because a\cdot b=|a||b|\cos \theta ]$
$=|a||b\times c|=2\times 6\sqrt{3}$
$=12\sqrt{3}$