Question

For the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin.
$3y+5=0$.

Answer 1

Given equation in vector form is $\overrightarrow{r}\cdot (0\hat{i}-1\cdot \hat{j}+0\cdot \hat{k})=\frac{5}{3}$ $[\because -y=\frac{5}{3}]$.

Clearly, $|0\hat{i}-1\cdot \hat{j}+0\cdot \hat{k}|=\sqrt{0^2+(-1{)}^2+0^2}=1$

$\therefore \overrightarrow{r}\cdot \hat{n}=p$, where $\hat{n}=(0\hat{i}-1\cdot \hat{j}+0\hat{k})$ and $p=\frac{5}{3}$.

Direction cosines of $\hat{n}$ are $0,-1,0$ and $p=\frac{5}{3}$.