Question

The intercept made by the plane $\overrightarrow{r}.\overrightarrow{n}=q$ on the $x-$axis is

• A. $\frac{q}{\hat{i}.\overrightarrow{n}}$
• B. $\frac{\hat{i}.\overrightarrow{n}}{q}$
• C. $(\hat{i}.\overrightarrow{n})q$
• D. $\frac{q}{\left|\overrightarrow{n}\right|}$

Answer 1

The correct option is A $\frac{q}{\hat{i}.\overrightarrow{n}}$

The equation of the plane is given as $\overrightarrow{r}.\overrightarrow{n}=q$
This can be written in normal form as $\overrightarrow{r}.\hat{n}=\frac{q}{\left|\overrightarrow{n}\right|}$
Hence, the perpendicular distance of the plane from origin is $\frac{q}{\left|\overrightarrow{n}\right|}$
Suppose, $d$ is the intercept made by the plane on $x$-axis.
$\Rightarrow \overrightarrow{d}.\hat{n}=\frac{q}{\left|\overrightarrow{n}\right|}$
$\Rightarrow d(\hat{i}.\hat{n})=\frac{q}{\left|\overrightarrow{n}\right|}$ ( $\because \overrightarrow{d}=d\hat{i}$)
$\Rightarrow d=\frac{q}{(\hat{i}.\hat{n})\left|\overrightarrow{n}\right|}$
$\Rightarrow d=\frac{q}{\hat{i}.\overrightarrow{n}}$