Three Dimensional Geometry — Class 12 Maths Solution

Since, DC's of line $OA$ are $\frac{a}{{\sqrt {{a^2} + {b^2} + {c^2}} }},\frac{b}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$

and $\frac{c}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$.

Also, $\quad \overrightarrow {\rm{n}} = \overrightarrow {{\rm{OA}}}$

$= \overrightarrow {\rm{a}} = a\widehat {\bf{i}} + b\widehat {\bf{j}} + c\widehat {\bf{k}}$

The equation of plane passes through $(a,b,c)$ and perpendicular to $OA$ is given by

$[\overrightarrow {\bf{r}} - \overrightarrow {\bf{a}} ] \cdot \overrightarrow {\bf{n}} = 0$

$\Rightarrow \overrightarrow r \cdot \overrightarrow n = \overrightarrow a \cdot \overrightarrow n$

$\Rightarrow [(x\widehat {\bf{i}} + y\widehat {\bf{j}} + z\widehat {\bf{k}}) \cdot (a\widehat {\bf{i}} + b\widehat {\bf{j}} + c\widehat {\bf{k}})] = (a\widehat {\bf{i}} + b\widehat {\bf{j}} + c\widehat {\bf{k}}) \cdot (a\widehat {\bf{i}} + b\widehat {\bf{j}} + c\widehat {\bf{k}})$

$\Rightarrow ax + by + cz = {a^2} + {b^2} + {c^2}$