Symmetries in space

In video game design software, an object located at point $P(4, -2, 6)$ must be reflected with respect to a control point (centre of symmetry) located at $M(1, 3, 2)$. Calculate the coordinates of the symmetric point $P'$ where the reflected object will appear.

It is known that the point $A'(0, 8, -4)$ is the symmetric point of $A(2, -2, 6)$ with respect to a centre of symmetry $C$. Determine the coordinates of this centre $C$ using the midpoint relationship.

Calculate the symmetric point of $Q(3, -5, 7)$ with respect to the coordinate plane $XY$.

Hint: Remember that the equation of the plane $XY$ is $z = 0$.

Given the point $P(1, 0, 1)$, find its symmetric point with respect to the plane $\pi: x + y + z = 5$ by following these steps:

1. Find the line $r$ perpendicular to $\pi$ passing through $P$.

2. Calculate the point of intersection $M$ between the line $r$ and the plane $\pi$ (foot of the perpendicular).

3. Use $M$ as the midpoint to find the symmetric point $P'$.

Calculate the coordinates of the point $Q$, which is the symmetric point of the point $P(1, 0, 5)$ with respect to the plane with equation $\pi: 2x + y - z - 3 = 0$. Briefly explain the steps followed (obtaining the perpendicular line, intersection point and midpoint).

Find the point symmetric to $P(1, 2, 3)$ with respect to the line $r$ that coincides with the $OX$-axis.

Calculate the symmetric point of $B(5, 5, 5)$ with respect to the line $s$ given by:

Note: This is a vertical line, parallel to the $OZ$ axis, that passes through the point $(1, 2, 0)$.

The figure shows the view from above (the $XY$ plane): the line $s$, being vertical, looks like a point, and the projection of $B$ is also shown.

Given the line $r$ defined by the parametric equations:

$r: \begin{cases} x = 1 + 2\lambda \\ y = \lambda \\ z = -1 + \lambda \end{cases}$

Find the point symmetric to $A(3, 2, 1)$ with respect to that line.

A laser beam is projected following the path of the line $r: \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{1}$. The ray strikes a polished metal surface located in the plane $XY$ (plane $z=0$). Calculate the equation of the line $s$ containing the reflected ray.

Note: The reflected line is the reflection of the incident line with respect to the plane of incidence.

Consider the points $A(1, 1, 1)$ and $B(3, 5, 5)$. Find the equation of the plane $\pi$ acting as a "plane of symmetry" between both points (that is, the plane with respect to which $A$ and $B$ are symmetric).