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Q.
The reflection of the point $\left(\alpha, \beta, \gamma\right)$ in the $xy$-plane is
Three Dimensional Geometry
Solution:
Projection of $P\left(\alpha, \beta, \gamma\right) xy$-plane is $Q\left(\alpha, \beta, 0\right)$
If $P'\left(\alpha', \beta', \gamma'\right)$ is the reflection of $P$ in $xy$-plane, then $Q$ is the mid-point of $PP'$
$\Rightarrow \left(\alpha, \beta, 0\right)=\left(\frac{\alpha+\alpha'}{2}, \frac{\beta+\beta'}{2}, \frac{\gamma+\gamma'}{2}\right)$
$\Rightarrow \frac{\alpha+\alpha'}{2}=\alpha$, $\frac{\beta+\beta'}{2}=\beta$, $\frac{\gamma+\gamma'}{2}=0$
$\Rightarrow \alpha'=\alpha, \beta'=\beta, \gamma'=-\gamma$
$\therefore $ Required reflection is $\left(\alpha, \beta, -\gamma\right)$