Exercises: 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 (center 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 center of symmetry $C$. Determine the coordinates of this center $C$ 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 $z = 0$.

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

A laser beam leaves the point $A(2, 3, 4)$ and reflects off a flat mirror whose equation is $\pi: y = 5$. Determine the coordinates of the point $A'$ symmetric with respect to that plane.

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 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.

In a crystal with a parallelogram structure, three vertices are known: $A(1, 1, 1)$, $B(3, 0, 2)$ and $C(4, 5, 6)$. If the center of the parallelogram $M$ is the center of symmetry connecting opposite vertices, find the fourth vertex $D$ knowing that it is symmetric to $B$ with respect to the midpoint of the segment $AC$.

An underwater drone is at position $D(4, -2, -5)$ (the XY plane represents sea level). First, it ascends vertically a certain distance by applying the translation vector $\vec{v} = (0, 0, 5)$. Next, it emits a sonar pulse that is specularly reflected off the seabed, which is completely flat and defined by the equation $z = -10$. What are the coordinates of the point symmetric to the drone's new position with respect to the seabed?

An architect is designing a metallic structure and needs to place two supports at points that are symmetric with respect to the center of a main beam. If one of the supports must be at point $P(3, -2, 4)$ and the center of the beam is at $M(1, 2, 1)$, determine the coordinates where the second support $P'$ should be placed.

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).

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.

In a virtual reality studio, an object located at $V(2, 4, 6)$ is reflected in a flat mirror that coincides with the coordinate plane $OXY$ ($z=0$).

1. a) Determine the symmetric point $V'$.
2. b) If the mirror is moved parallelly to position $\pi: z = 2$, what would be the new symmetric point $V''$?

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).

It is known that the point $P'(5, -3, 0)$ is the symmetric point of $P(1, 1, 0)$ with respect to a line $r$. If the direction vector of the line is $\vec{v} = (1, 1, 1)$, determine the continuous equation of the line $r$.

A laser ray leaves the point $A(1, 2, 3)$, bounces off a flat pane of glass whose equation is $\pi: x + y + z = 12$ and reaches the point $B$. Knowing that the point where the laser touches the glass is the point of minimum incidence (the midpoint of the path if $A$ and $B$ were symmetric), find the coordinates of $B$ knowing that it is symmetric to $A$ with respect to the plane $\pi$.

Determine the value of the parameters $a$ and $b$ so that the points $P(1, 2, -1)$ and $Q(a, b, 3)$ are symmetric with respect to the point $C(0, 1, 1)$. Once found, check whether the vector $\vec{PQ}$ is parallel to the vector $\vec{u} = (1, 1, -4)$.

A surveillance drone is located at the point $A(1, -2, 3)$. To avoid radar detection, it must move to the point $B(5, 4, 1)$, making a technical stopover on a flat platform represented by the plane $\pi: x - y + 2z - 2 = 0$. Determine the coordinates of a point $P$ on that plane so that the total path $AP + PB$ is minimized.

Hint: Use the reflection of the point $A$ across the plane $\pi$ to solve this geometric optimization problem.

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.

Determine the value of the parameter $k \in \mathbb{R}$ so that the point symmetric to $P(1, k, 2)$ with respect to the line $r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 \lambda \end{cases}$ belongs to the plane $\pi: 3x - y + z - 10 = 0$.

Consider the points $A(2, 0, 1)$ and $B(0, 2, -1)$. It is known that the point $B$ is the reflection of $A$ in a line $r$ passing through the origin of coordinates. Find the implicit equations of that line $r$.

Find the equation of the plane $\pi'$ that is symmetric to the plane $\pi: 2x - y + z - 4 = 0$ with respect to the plane of symmetry $\sigma: x + y + z - 1 = 0$.

Hint: Analyse the relative position of both planes. If they intersect, the line of intersection will belong to the required plane.

Let the lines be $r: \begin{cases} x + y - z = 1 \\ 2x - y = 0 \end{cases}$ and $s: \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$. Calculate the symmetric point of $Q(0, 0, 10)$ with respect to the plane containing both lines. If no such plane exists, explain why and calculate the symmetric point of $Q$ with respect to the line $s$.

Given the point $A(1, 2, 3)$, calculate the coordinates of its symmetric points with respect to the three coordinate planes ($XY$, $XZ$, and $YZ$). Determine whether the four points (the original one and the three symmetric points) are coplanar and, if so, find the equation of the plane that contains them.

En un proyecto de arquitectura se desea construir una estructura simétrica respecto a un pilar central representado por la recta $r: x-1 = y = \dfrac{z+2}{2}$. Uno de los vértices de la estructura está en el punto $V(3, 1, 0)$.

1. a) Find the symmetric vertex $V'$ of $V$ with respect to the pillar $r$.

1. b) Calcula el volumen del tetraedro cuyos vértices son $V$, $V'$, el origen de coordenadas $O(0,0,0)$ y el punto $A(1, 0, -2)$ de la recta $r$.