Question

Find the angles at which the normal vector to the plane $4x+8y+z=5$ is inclined to the coordinate axes.

Answer 1

Let $\overrightarrow{n}$ be a vector normal to the plane. Since direction ratios of normal to the plane are proportional to 4, 8, 1.

Therefore, $\overrightarrow{n}=4\hat{i}+8\hat{j}+\hat{k}$

Direction cosines of $\overrightarrow{n}$ are

$\frac{4}{9},\frac{8}{9},\frac{1}{9}$

Let $\alpha ,\beta ,\gamma$ be the angles made by $\overrightarrow{r}$ with x-axis, y-axis and z-axis respectively.

Then,

$\alpha =co{s}^{-1}\left(\frac{4}{9}\right),\beta =co{s}^{-1}\left(\frac{8}{9}\right),\gamma =co{s}^{-1}\left(\frac{1}{9}\right)$