Question

Let $a=2i+j-2k$ and $b=i+j$. If $c$ is a vector such that $a.c=\left|c\right|,|c-a|=2\sqrt{2}$ and the angle between $a\times b$ and $c$ is ${30}^0$ then $|(a\times b)\times c|$ is equal to

• A. $\frac{2}{3}$
• B. $\frac{3}{2}$
• C. $2$
• D. $3$

Answer 1

The correct option is B $\frac{3}{2}$

We have, $a\times b=2i-2j+k$

$\therefore |(a\times b)\times c|=|a\times b|\left|c\right|\sin {30}^0$

$=\sqrt{2^2+{(-2)}^2+1^2}\left|c\right|\frac{1}{2}=\frac{3}{2}\left|c\right|$

Given $|c-a|=2\sqrt{2}$

$\Rightarrow {(c-a)}^2=9\Rightarrow {\left|c\right|}^2+{\left|a\right|}^2-2c.a=8$

$\Rightarrow {\left|c\right|}^2-2\left|c\right|+1=0{(\left|c\right|-1)}^2=0\Rightarrow \left|c\right|=1$

$\therefore |(a\times b)\times c|=\frac{3}{2}$