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 ${30}^o$, 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 $\frac{3}{2}$

$a\times b=\left|\begin{array}{ccc}i& j& k\\ 2& 1& -2\\ 1& 1& 0\end{array}\right|=2i-2j+k$
$\therefore |a\times b|=\sqrt{4+4+1}=3$
$|c-a|=2\sqrt{2}$

$\Rightarrow {|c-a|}^2$
$={(c-a)}^2=8$
$\Rightarrow {\left|c\right|}^2-2a.c+{\left|a\right|}^2=8$
$\Rightarrow {\left|c\right|}^2-2\left|c\right|+9=8$
$\Rightarrow {\left|c\right|}^2-2\left|c\right|+1\Rightarrow \left|c\right|=1$
$\therefore |(a\times b)\times c|=\left|\right(a\times b\left)c\right|\sin {30}^o=\frac{3}{2}$