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 (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 $z = 0$.

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 centre of the parallelogram $M$ is the centre 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?

Lucía needs to find the coordinates of the symmetric point of $P(1, 0, -2)$ with respect to the plane $\pi: 2x - y + z + 5 = 0$. On her exam, she writes the following:

1. Write the line $r$ perpendicular to $\pi$ passing through $P$: $r: (x, y, z) = (1 + 2t, -t, -2 + t)$.

2. Substitute the parametric equations of $r$ into the plane $\pi$ to find the intersection:

1. Substitute the parameter $t$ into the line to obtain the point $M\left(-\frac{2}{3}, \frac{5}{6}, -\frac{17}{6}\right)$.

2. Conclude: "The desired symmetric point is $P'\left(-\frac{2}{3}, \frac{5}{6}, -\frac{17}{6}\right)$".

1. a) Identify Lucía's conceptual error and explain, using the distance of the points to the plane $\pi$, why her result is geometrically impossible.
2. b) Correct the procedure and calculate the correct coordinates of the symmetric point $P'$.

Consider the point $P(4, 2, -1)$ and the plane $\pi: x + y - 2 = 0$.

1. a) Before performing detailed analytical calculations, look at the normal vector of the plane $\pi$ and answer with reasoning: which coordinate of $P$ will remain invariant in its symmetric point $P'$? Furthermore, taking into account that the initial coordinates of $P$ satisfy $x + y = 6 > 2$, estimate with reasoning whether the $x$ and $y$ coordinates of $P'$ should be greater or smaller than those of $P$.
2. b) Analytically determine the orthogonal projection $M$ of $P$ onto $\pi$ and find the coordinates of the symmetric point $P'$ by applying the midpoint formula.
3. c) Check whether the numerical values obtained for $P'$ confirm the geometric estimates you made in part a).

An architect is designing a metallic structure and needs to place two supports at points that are symmetric with respect to the centre of a main beam. If one of the supports must be at point $P(3, -2, 4)$ and the centre 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 $\vec{u} = (1, 1, -4)$.

It is known that the point $P'(-1, 5, 2)$ is the symmetric point of $P(3, 1, -2)$ with respect to an unknown reflection plane $\pi$.

1. a) Deduce with reasoning the general equation of the plane $\pi$. Justify each step by explaining which two fundamental geometric properties relate the segment $PP'$ to the plane of symmetry.
2. b) A line in space is globally invariant under a specular reflection (that is, its symmetric line coincides with itself, $r' = r$) if and only if it satisfies one of two possible geometric conditions with respect to the plane $\pi$. State with reasoning what those two conditions are, and determine whether the line $r: \frac{x - 1}{2} = \frac{y - 3}{1} = \frac{z}{1}$ is invariant with respect to the plane $\pi$ found in the previous section.

In the modernization project of the M-30 tunnel in Madrid, a three-dimensional coordinate system (in meters) is used to place symmetric acoustic sensors. The left side wall of the section is modeled by the plane $\pi: 2x - y + 2z - 12 = 0$. On the opposite wall, a first sensor has been fixed at the point $P(7, 1, 4)$.

1. a) For calibration and acoustic balance purposes, a second sensor must be placed at position $P'$, symmetric to $P$ with respect to the wall $\pi$. Determine with reasoning the coordinates where $P'$ should be anchored.
2. b) A data cable connects the sensor $P$ to a junction box located at $C(-2, 1, 4)$. The technical team proposes locating the corresponding symmetric box at the point $C^*(2, -1, 5)$. Without having to calculate the entire symmetric line, check whether $C^*$ is exactly the symmetric point of $C$ with respect to $\pi$. If not, identify which quick geometric check (the perpendicular direction or the position of the midpoint) reveals the error, and calculate the correct position $C'$.

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 minimised.

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

To find the symmetric line $s'$ of a line $s$ with respect to a plane $\pi$, two strategies are usually compared:

• Method A: Find the intersection point $I = s \cap \pi$ (which, being on the plane, is invariant, $I' = I$), calculate the symmetric point $A'$ of any other point $A \in s$ with respect to $\pi$, and draw the line passing through $I$ and $A'$.

• Method B: Select two distinct points $A, B \in s$, independently calculate their symmetric points $A'$ and $B'$ with respect to $\pi$, and write the line connecting $A'$ and $B'$.

1. a) Critically analyze both approaches: for which relative position of the line $s$ with respect to the plane $\pi$ is Method A unfeasible? How does the geometry of the problem allow the calculation to be simplified in that particular case without having to reflect two full points as the standard Method B would require?
2. b) Given the line $s: \frac{x - 1}{1} = \frac{y + 1}{-1} = \frac{z - 2}{1}$ and the plane $\pi: x + 2y + z + 5 = 0$, analyze their relative position, justify which is the most efficient solution method, and obtain the symmetric equations of the symmetric line $s'$.

In three-dimensional space, consider the points $P(2, 3, 1)$ and $P'(-2, 1, 5)$, which are symmetric with respect to an axis of symmetry (a line $e$).

1. a) Unlike what happens in the plane $\mathbb{R}^2$, where two symmetric points determine a unique axis of symmetry, in $\mathbb{R}^3$ there are infinitely many lines that can act as an axis of symmetry for the pair $P$ and $P'$. Explain with reasoning why this happens and describe what geometric locus is formed by all possible axes of symmetry associated with these points.
2. b) The constraint that the axis of symmetry $e$ must be contained in the plane $\pi: x + y + z - 5 = 0$ is added. Justify why this condition is geometrically compatible with the positions of $P$ and $P'$, and obtain the symmetric equations of the axis of symmetry $e$.
3. c) Analytically verify the validity of the obtained line $e$ by checking two essential metric conditions: that the midpoint of $PP'$ belongs to the line $e$, and the orthogonality between the direction vector of $e$ and the vector $\vec{PP'}$.