Find the equation of the plane through the intersection of the planes $\overrightarrow {\rm{r}} \cdot (\widehat {\rm{i}} + 3\widehat {\rm{j}}) - 6 = 0$ and $\overrightarrow {\rm{r}} \cdot (3\widehat {\rm{i}} - \widehat {\rm{j}} - 4\widehat {\rm{k}}) = 0,$ whose perpendicular distance from origin is unity.

It is given that, $\overrightarrow {{{\rm{n}}_1}} = (\widehat {\rm{i}} + 3\widehat {\rm{j}}),{d_1} = 6$

and $\overrightarrow {{{\rm{n}}_2}} = (3\widehat {\rm{i}} - \widehat {\rm{j}} - 4\widehat {\rm{k}}),{d_2} = 0$

Using the relation, $\overrightarrow {\rm{r}} \cdot \left( {\overrightarrow {{{\rm{n}}_1}} + \lambda \overrightarrow {{{\rm{n}}_2}} } \right) = {d_1} + {d_2}\lambda$

$\Rightarrow$ $\overrightarrow {\rm{r}} \cdot [(\widehat {\rm{i}} + 3\widehat {\rm{j}}) + \lambda (3\widehat {\rm{i}} - \widehat {\rm{j}} - 4\widehat {\rm{k}})] = 6 + 0 \cdot \lambda$

$\Rightarrow$ $\overrightarrow {\rm{r}} \cdot [(1 + 3\lambda )\widehat {\rm{i}} + (3 - \lambda )\widehat {\rm{j}} + \widehat {\rm{k}}( - 4\lambda )] = 6$

……(i)
On dviding both sides by $\sqrt {{{(1 + 3\lambda )}^2} + {{(3 - \lambda )}^2} + {{( - 4\lambda )}^2}}$,

we get
$\frac{{\overrightarrow {\rm{r}} \cdot [(1 + 3\lambda )\widehat {\rm{i}} + (3 - \lambda )\widehat {\rm{j}} + \widehat {\rm{k}}( - 4\lambda )]}}{{\sqrt {{{(1 + 3\lambda )}^2} + {{(3 - \lambda )}^2} + {{( - 4\lambda )}^2}} }}$

$= \frac{6}{{\sqrt {{{(1 + 3\lambda )}^2} + {{(3 - \lambda )}^2} + {{( - 4\lambda )}^2}} }}$

Since, the perpendicular distance from origin is unity.

$\therefore \frac{6}{{\sqrt {{{(1 + 3\lambda )}^2} + {{(3 - \lambda )}^2} + {{( - 4\lambda )}^2}} }} = 1$

$\Rightarrow$ ${(1 + 3\lambda )^2} + {(3 - \lambda )^2} + {( - 4\lambda )^2} = 36$

$\Rightarrow$ $1 + 9{\lambda ^2} + 6\lambda + 9 + {\lambda ^2} - 6\lambda + 16{\lambda ^2} = 36$

$\Rightarrow$ $26{\lambda ^2} + 10 = 36$
$\Rightarrow$ ${\lambda ^2} = 1$

$\therefore \lambda = \pm 1$

Using Eq. (i), the required equation of plane is $\overrightarrow {\rm{r}} \cdot [(1 \pm 3)\widehat {\rm{i}} + (3 \mp 1)\widehat {\rm{j}} + ( \mp 4)\widehat {\rm{k}}] = 6$

$\Rightarrow$ $\overrightarrow {\rm{r}} \cdot [(1 + 3)\widehat {\rm{i}} + (3 - 1)\widehat {\rm{j}} + ( - 4)\widehat {\rm{k}}] = 6$

and $\overrightarrow {\rm{r}} \cdot [(1 - 3)\widehat {\rm{i}} + (3 + 1)\widehat {\rm{j}} + 4\widehat {\rm{k}}] = 6$

$\Rightarrow$ $\overrightarrow {\rm{r}} \cdot (4\widehat {\rm{i}} + 2\widehat {\rm{j}} - 4\widehat {\rm{k}}) = 6$

and $\overrightarrow {\rm{r}} \cdot ( - 2\widehat {\rm{i}} + 4\widehat {\rm{j}} + 4\widehat {\rm{k}}) = 6$

$\Rightarrow$ $4x + 2y - 4z - 6 = 0$
and $- 2x + 4y + 4z - 6 = 0$