Question

The reflection of the point $\overrightarrow{a}$ in the plane $\overrightarrow{r}\cdot \overrightarrow{n}=q$ is-

• A. $\overrightarrow{a}+\frac{(\overrightarrow{q}-\overrightarrow{a}\cdot \overrightarrow{n})}{\mid \overrightarrow{n}\mid }$
• B. $\overrightarrow{a}+2\left(\frac{(\overrightarrow{q}-\overrightarrow{a}\cdot \overrightarrow{n})}{\mid {\overrightarrow{n}}^2\mid }\right)\overrightarrow{n}$
• C. $\overrightarrow{a}+\frac{2(\overrightarrow{q}+\overrightarrow{a}\cdot \overrightarrow{n})}{\mid {\overrightarrow{n}}^2\mid }\overrightarrow{n}$
• D. None of these.

Answer 1

Answer: (B)

$\overrightarrow{a}+2\left(\frac{(\overrightarrow{q}-\overrightarrow{a}\cdot \overrightarrow{n})}{\mid {\overrightarrow{n}}^2\mid }\right)\overrightarrow{n}$

The equation of line passing through $\overrightarrow{a}$ and perpendicular to plane be $\overrightarrow{r}.\overrightarrow{n}=q$ is $\overrightarrow{r}=\overrightarrow{a}+t\overrightarrow{n}$
Let point of intersection of the plane with the line be $P$: $\overrightarrow{a}+t\overrightarrow{n}$
Since, $P$ lies on the plane $\overrightarrow{r}.\overrightarrow{n}=q$
Therefore, $(\overrightarrow{a}+t\overrightarrow{n}).\overrightarrow{n}=q$
$\Rightarrow t=\frac{q-\overrightarrow{a}.\overrightarrow{n}}{|n{|}^2}$
Let image be $\overrightarrow{b}$
Since $P$ is the mid point of $\overrightarrow{a}$ and $\overrightarrow{b}$
Therefore, $\overrightarrow{a}+t\overrightarrow{n}=\frac{\overrightarrow{a}+\overrightarrow{b}}{2}$
$\Rightarrow \overrightarrow{b}=\overrightarrow{a}-2t\overrightarrow{n}=\overrightarrow{a}+2\left(\frac{(\overrightarrow{q}-\overrightarrow{a}\cdot \overrightarrow{n})}{{\mid \overrightarrow{n}\mid }^2}\right)\overrightarrow{n}$