Let ${\pi }_1$ be the plane passing through the point $2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and perpendicular to the vector $a\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}$ and ${\pi }_2$ be the plane passing through the point $\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and perpendicular to the vector $\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$. If $\theta$ is the angle between the planes ${\pi }_1$ and ${\pi }_2$ and $\cos \theta =-\sqrt{\frac{3}{7}}$, then the integral value of $a$ is

Given,

${\pi }_1$ be the plane passing through the point $2\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and perpendicular to the vector $a\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}$

So normal vector ${\overrightarrow{n}}_1=a\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}$

And ${\pi }_2$ be the plane passing through the point $\hat{\mathrm{i}}+2\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and perpendicular to the vector $\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$,

So nomal vector to the plane will be ${\overrightarrow{n}}_2=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}$

Now $\theta$ is the angle between the planes ${\pi }_1$ and ${\pi }_2$ is given by,

$\cos \theta =-\sqrt{\frac{3}{7}}$

$\Rightarrow \left|\frac{{\overrightarrow{n}}_1.{\overrightarrow{n}}_2}{\left|{\overrightarrow{n}}_1\right|.\left|{\overrightarrow{n}}_2\right|}\right|=-\sqrt{\frac{3}{7}}$

$\Rightarrow \left|\frac{\left(a\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right).\left(\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)}{\left|a\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right|.\left|\hat{\mathrm{i}}-2\hat{\mathrm{j}}+\hat{\mathrm{k}}\right|}\right|=-\sqrt{\frac{3}{7}}$

$\Rightarrow \left|\frac{a-4-3}{\sqrt{a^2+4+9}.\sqrt{1^2+2^2+1^2}}\right|=-\sqrt{\frac{3}{7}}$

$\Rightarrow \left|\frac{a-7}{\sqrt{a^2+13}.\sqrt{6}}\right|=-\sqrt{\frac{3}{7}}$

$\Rightarrow \frac{{\left(a-7\right)}^2}{\left(a^2+13\right)\times 6}=\frac{3}{7}$

Then the integral value of $a$ from options is $1$