Question

Equation of a plane is ${\displaystyle \overrightarrow{r}(3\hat{i}-4\hat{j}+12\hat{k})=8}$. Find the length of the perpendicular from the origin to the plane.

Answer 1

${\displaystyle \overrightarrow{r}(3\hat{i}-4\hat{j}+12\hat{k})=8..........\left(i\right)}$

${\displaystyle \overrightarrow{n}=3\hat{i}-4\hat{j}+12\hat{k}}$

${\displaystyle \left|\overrightarrow{n}\right|=\sqrt{9+16+144}=13}$

${\displaystyle \therefore \overrightarrow{n}=\frac{3\hat{i}-4\hat{j}+12\hat{k}}{13}}$

${\displaystyle \therefore \overrightarrow{r}.\left(\frac{3\hat{i}-4\hat{j}+12\hat{k}}{13}\right)=\frac{8}{13}............by\left(i\right)}$

${\displaystyle \overrightarrow{r}.\overrightarrow{n}=p}$

∴ Perpendicular distance from the origin is ${\displaystyle \frac{8}{13}}$