Question

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

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

Answer 1

Answer: (B)

Given,
$a \cdot b=0$
$a \cdot c=0$
$\therefore a \| b \times c$
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} $