Let $a= 4i - j +3k , b =- 2i + j - 2k $
and $c = xi + yj + zk $
Given, $a \cdot c =0$
i.e., $4x-y+3z=0\, \dots(i)$
and $b \cdot c =0$
i.e., $-2 x+y-2 z=0\, \dots(ii)$
Also, $| c |=9$
i.e., $x^{2}+y^{2}+z^{2}=81\, \dots(iii)$
Now, from Eqs. (i) and (ii), we get
$2 x+z=0$
$ \Rightarrow z=-2 x$
On putting this value in Eq. (iii), we get
$x^{2}+y^{2}+4 x^{2}=81$
$\Rightarrow 5 x^{2}+y^{2}=81\, \dots(iv)$
On multiplying Eq. (i) by 2 and Eq. (ii) by 3 and then adding, we get
$8x-2y+6 z=0 $
$-6 x+3 y-6 z=0$
______________
$2x+y=0$
$\Rightarrow y=-2x$
On putting this value in Eq. (iv), we get
$5 x^{2}+4 x^{2}=81$
$\Rightarrow 9 x^{2}=81$
$\Rightarrow x^{2}=9$
$\Rightarrow x=\pm 3$
$\therefore y=\mp 6$ and $z=\mp 6$
$\therefore $ Required vector, $c =xi+yj +zk$
$=\pm 3i \mp 6j \mp 6k $
$=3i-6 j-6 k$
or
$=-3i+6j+6k$