Question

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

Answer 1

Equation of plane is $\overline{r}.(3\hat{i}-4\hat{j}+12\hat{k})=8$
compare $\overrightarrow{r}.\overrightarrow{n}=p$
Then $\overrightarrow{n}=3\hat{i}-4\hat{j}+12\hat{k},p=8$
$\therefore$ Length of the perpendicular from the origin to the plane is
$d=\frac{p}{\left|\overrightarrow{n}\right|}=\frac{8}{\sqrt{3^2+{(-4)}^2+{12}^2}}$
$=\frac{8}{\sqrt{9+16+144}}=\frac{8}{\sqrt{169}}=\frac{8}{13}$ units