Question

If $a=i+2j+3k$ and $b=i\times (a\times i)+j\times (a\times j)+k\times (a\times k)$ then length of b is equal to :

• A. $\sqrt{12}$
• B. $2\sqrt{12}$
• C. $3\sqrt{14}$
• D. $2\sqrt{14}$

Answer 1

The correct option is D $2\sqrt{14}$

We have, $a=i+2j+3k$

$\therefore b=i\times (a\times i)+j\times (a\times j)+k\times (a\times k)$ ...(i)

Now, $i\times (a\times i)=(i.i)a-(i.a)i$
$=1(i+2j+3k)-\left(1\right)i$
$=2j+3k$

Similarly $j(a\times j)=i+3k$
and $k\times (a\times k)=i+2j$
$\therefore$ From Eq. (i),

$b=2j+3k+i+3k+i+2j$

$=2i+4j+6k$

$\Rightarrow |b|=\sqrt{4+16+36}=2\sqrt{14}$