Let a vector $\overline{a}$ be coplanar with vectors $\overline{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\overline{c}=\hat{i}-\hat{j}+\hat{k}$ . If $\overline{a}$ is $\perp r$ to $\overline{d}=3\hat{i}+2\hat{j}+6\hat{k}$ and $\left|\overline{a}\right|=\sqrt{10}$ then possible value of  $\left[\begin{array}{ccc}\overline{a}& \overline{b}& \overline{c}\end{array}\right]+\left[\begin{array}{ccc}\overline{a}& \overline{b}& \overline{d}\end{array}\right]+\left[\begin{array}{ccc}\overline{a}& \overline{c}& \overline{d}\end{array}\right]$  is

$\text{ Let }\overline{a}=xi+yj+zk$

$\left[\begin{array}{l}\overline{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{c}\end{array}\right]=0\Rightarrow 2x-y-3z=0\dots .\left(1\right)$

$\overline{a}\cdot \overline{d}=0\Rightarrow 3x+2y+6z=0\dots .\left(.2\right)$

$\text{ From }1\&2,x=0,y=21t,z=-7t$

$x^2+y^2+z^2=10\Rightarrow t=\pm \frac{1}{7}$

$\therefore \overline{a}=-3\hat{J}+\hat{k}\text{ (or) }3\hat{J}-\hat{k}$

$\left[\overline{a}\overline{b}\overline{c}\right]+\left[\overline{a}\overline{b}\overline{d}\right]+\left[\begin{array}{l}\overline{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{d}\end{array}\right]=\left[\overline{a}\overline{b}\overline{c}\right]+[\overline{a}\overline{b}+\overline{c}\overline{d}]$

$=42\left(\text{ or }\right)-42$