Exercises: Parallelism and perpendicularity between lines and planes in space

Determine the relative position of the line $r$ and the plane $\pi$ by checking the relationship between their characteristic vectors. The line is given by the continuous equation $r: \dfrac{x-1}{2} = \dfrac{y+3}{-1} = \dfrac{z}{4}$ and the plane by the general equation $\pi: 2x - y + 4z - 5 = 0$.

1. a) Write a direction vector $\vec{v}_r$ of the line and a normal vector $\vec{n}$ of the plane.

1. b) Is the direction vector of the line parallel to the normal vector of the plane? Justify your answer.

1. c) Justify what geometric implication this fact has for the relative position of $r$ and $\pi$.

A robotic arm in an assembly plant moves along a straight path $s$. The path passes through the control point $A(2, -1, 5)$ and must be perpendicular to the support panel, which is represented by the plane $\pi: x + 3y - 2z + 4 = 0$. Find the parametric equations of the line $s$ followed by the robotic arm.

Check whether the plane $\pi_1: 2x - 4y + 2z - 7 = 0$ and the plane $\pi_2: x - 2y + z + 5 = 0$ are parallel. To do this, analyse whether their normal vectors are proportional. If they are parallel, reason whether they are the same plane (coincident) or if they have no points in common.

Given the point $Q(3, -1, 4)$ and the plane $\alpha: 2x + y = 0$, find the equation of the line $r$ passing through $Q$ and perpendicular to the plane $\alpha$. Take special care when identifying the coordinates of the normal vector of the plane, paying attention to the absence of the variable $z$.

Find the value of the parameter $k$ so that the line $r: \dfrac{x-2}{3} = \dfrac{y}{-2} = \dfrac{z+1}{1}$ is parallel to the plane $\pi: kx + 3y + z - 7 = 0$. Remember that for a line and a plane to be parallel, the scalar product of their characteristic vectors must be equal to zero.

In a solar energy project, a flat panel represented by the plane $\pi \equiv 2x + y - z = 4$ is installed. A support mast is defined by the line $r \equiv \{x = 1+t, y = 2-t, z = 3t\}$. At midday, the sun's rays strike perpendicularly to the panel. Calculate the equation of the line $s$ representing the orthogonal projection of the mast $r$ onto the solar panel $\pi$ (its "shadow" under perpendicular incidence). Also determine the point where the mast passes through the panel.

Consider the line $r$ passing through the points $A(1, 0, 2)$ and $B(2, 1, 0)$. Find the coordinates of a point $P$ lying on the plane $\pi \equiv x + y + z = 10$, such that the segment connecting point $P$ with its orthogonal projection onto the line $r$ is perpendicular to the plane $\pi$.

Let the family of planes be $\pi_a \equiv ax + y + z = a$. Determine the value of $a$ such that the orthogonal projection of the line $r \equiv \frac{x-1}{1} = \frac{y}{2} = \frac{z+1}{-1}$ onto the plane $\pi_a$ is a line parallel to the coordinate plane $YZ$. Once $a$ is found, write the equation of said projection line in parametric form.

An architect designs a pyramidal structure with vertex at $V(2, 2, 6)$ and whose base is contained in the plane $\pi \equiv x + 2y + 2z = 6$. Find the foot of the altitude of the pyramid (orthogonal projection of $V$ onto $\pi$) and determine the volume of the tetrahedron formed by the origin of coordinates and the points of intersection of the plane $\pi$ with the three Cartesian axes.

Álvaro has to classify the relative position between the line $r: \frac{x-1}{2} = \frac{y+3}{-1} = \frac{z}{1}$ and the plane $\pi: 2x - y + z + 4 = 0$. In his solution, he writes the following:

“The direction vector of the line is $\vec{v} = (2, -1, 1)$ and the normal vector of the plane is $\vec{n} = (2, -1, 1)$. Since $\vec{v} = \vec{n}$, both vectors are proportional, which proves that the line and the plane are parallel”.

1. a) Explain Álvaro's conceptual error. Describe the geometric relationship that a line and a plane actually hold when their direction and normal vectors are proportional, relying on the meaning of the normal vector to a flat surface.
2. b) Correct the exercise: determine the true relative position between $r$ and $\pi$ and find the coordinates of their point of intersection, if it exists.

It is known that the orthogonal projection of the external point $P(3, -1, 2)$ onto an unknown plane $\pi$ is the point $Q(1, 1, 0)$.

1. a) Justify geometrically why the vector $\vec{QP}$ constitutes a normal vector to the plane $\pi$ and write the general equation of that plane.
2. b) Write the continuous equation of a line $r$ that passes through point $P$ and is strictly parallel to the plane $\pi$. How many distinct lines satisfy these two conditions simultaneously? Justify your answer by describing what geometric figure all those lines form in space.

An architect is designing a ramp modelled as a plane $\pi$ with general equation $x + y + z - 6 = 0$. To install a lighting sensor, they need to know the orthogonal projection of the point $P(1, 2, 5)$ onto that plane. Calculate the coordinates of the point $Q$ representing this projection (called the foot of the perpendicular).

Find the orthogonal projection of the point $A(2, 0, -1)$ onto the line $r$ given by its equation in continuous form:

Hint: Remember that you can construct an auxiliary plane perpendicular to the line passing through the point $A$.

The line $r$, which passes through the points $A(1, 0, 0)$ and $B(1, 1, 1)$, is projected onto the plane $\pi: x = 0$.

1. Find the orthogonal projection of the points $A$ and $B$ onto the plane.

2. Write the equation of the projected line $r'$ passing through these new points.

An engineer is designing the path of a fibre optic cable that must be parallel to a protection panel. The panel is represented by the plane $\pi: 2x - 3y + az = 5$ and the path of the cable by the line $r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 + 2\lambda \end{cases}$.

1. a) Determine the value of the parameter $a \in \mathbb{R}$ so that the path of the cable is parallel to the panel.
2. b) For the value of $a$ found, check whether the cable lies within the panel or if there is a constant distance between them.

Calculate the equation of the plane $\alpha$ that passes through the point $P(1, 0, -2)$ and is perpendicular to the line $s$ defined by the intersection of the planes:

$s: \begin{cases} x + y - z = 4 \\ 2x - y + 3z = 1 \end{cases}$

Also determine the point of intersection between the line $s$ and the plane $\alpha$ found (orthogonal projection of $P$ onto $s$).

Given the line $r \equiv \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{3}$ and the plane $\pi: 3x + y - z = 4$, find the equation of the plane $\beta$ that satisfies simultaneously:

1. It contains the line $r$.

2. It is perpendicular to the plane $\pi$.

Consider the planes $\pi_1: x - 2y + z = 0$ and $\pi_2: 3x + y - z = 5$.

1. a) Determine the equation of the line $r$ resulting from the intersection of both planes.
2. b) Find the equation of a plane $\pi_3$ that passes through the point $Q(0, 0, 1)$ and is perpendicular to the line $r$.

An antenna is to be placed at a point $M$ on a flat hillside represented by the plane $\pi: x + 2y + 2z = 9$. The antenna must be as close as possible to a base station situated at the point $P(5, 4, 10)$.

1. a) Calculate the coordinates of the point $M$ (orthogonal projection of $P$ onto $\pi$).
2. b) If the antenna must be connected to the station by a straight cable, determine the parametric equations of the line followed by the cable.

Determine the relative position of the plane $\pi: mx + y - z = 1$ and the line $r \equiv \frac{x-1}{2} = \frac{y}{m} = \frac{z+3}{3}$ depending on the real parameter $m$. Specify for which values they are perpendicular and for which the line is parallel to the plane.

Two lines in space, $a$ and $b$, are skew lines. If the line $a$ is parallel to a horizontal plane $\pi$ and the line $b$ is perpendicular to that same plane $\pi$:

1. a) What angle do the directions of the lines $a$ and $b$ form?
2. b) Is it possible to find a plane containing both lines at the same time? Why?
3. c) Can we state that the two lines are perpendicular even if they do not have any point in common?

Observe the corner of a room where two walls and the floor meet. This set forms a trihedral angle at the vertex of the bottom corner.

1. a) What are the measures of the three plane angles that define the faces of this trihedron?
2. b) What is the sum of the measures of the three dihedral angles formed between the walls and the floor?
3. c) If a fly flies following a path parallel to the line of the skirting board of one wall, is that path also parallel to the plane of the other wall? Explain why.

Given the line $r: \frac{x-1}{2} = \frac{y-1}{1} = \frac{z}{-1}$ and the plane $\pi: x + y + z - 4 = 0$:

1. a) Verify that the line $r$ is not perpendicular to the plane $\pi$ and find their point of intersection $I$.
2. b) To determine the orthogonal projection of $r$ onto $\pi$, two students consider two different methods:

• Method A (point projection): since $I$ already belongs to the plane, it suffices to project orthogonally any other point $P \in r$ ($P \neq I$) to obtain $P'$, and the projected line will be the one passing through $I$ and $P'$.

• Method B (projecting plane): find the equation of the plane $\pi'$ containing $r$ and perpendicular to $\pi$, and express the projection as the intersection line $\pi' \cap \pi$.

Apply one of the two methods to calculate the projected line. Then, reason which method is computationally faster in this case and which of the two would still be applicable if the line $r$ were strictly parallel to the plane $\pi$.

Consider the line $r: \frac{x-1}{1} = \frac{y-2}{1} = \frac{z-1}{2}$ and the external point $P(1, 2, 7)$.

1. a) Estimate: Point $A(1, 2, 1)$ belongs to line $r$. Notice that $P$ is located exactly vertically above $A$, but at a higher altitude ($z = 7$). Knowing that the direction vector $\vec{v} = (1, 1, 2)$ moves forward toward positive values of $x$, $y$, and $z$ simultaneously, reason qualitatively before performing calculations:

• Whether the $z$-coordinate of the foot of the perpendicular $P'$ (orthogonal projection of $P$ onto $r$) will be strictly between $1$ and $7$.

• Whether the $x$- and $y$-coordinates of that point $P'$ will be greater or smaller than those of $A$.

1. b) Verify: Algebraically obtain the coordinates of $P'$ using the intersection of $r$ with the plane perpendicular to it that passes through $P$. Check whether the values obtained confirm your previous geometric deductions.

Calculate the coordinates of the point $Q$, which is the orthogonal projection of the point $P(3, -1, 5)$ onto the line $r$ defined by the following equation in continuous form:

Find the point $P'$ that represents the orthogonal projection of the point $P(1, 0, -3)$ onto the plane $\pi$ with equation:

A light source located at a very distant point casts rays parallel to the direction of the vector $\vec{v} = (0, 0, -1)$. If a steel cable follows the path of the line $r$, find the equation of the shadow it casts on a flat roof represented by $\pi$:

Obtain the implicit equation of the projecting plane of the line $r$ onto the plane $\pi$. Remember that the projecting plane is the plane perpendicular to $\pi$ that contains the line $r$.

A laser beam starts from point $P(1, -2, 3)$ and travels toward a plane mirror represented by the equation $\pi: 2x - y + z - 5 = 0$. The beam is reflected by the mirror following the laws of optical reflection (the incident ray, the normal to the plane at the point of incidence, and the reflected ray lie in the same plane, and the angles of incidence and reflection are equal).

Determine:

1. a) The coordinates of the point of incidence $Q$, knowing that the incident ray is perpendicular to the plane $\pi$.
2. b) The equation of the line containing the reflected ray if, instead of being perpendicular, the ray were to start from $P$ and strike the plane at point $A(1, 0, 3)$.
3. c) Justify with reasoning whether the incident ray in part a) and the normal vector of the plane are parallel or perpendicular.

In the design of a metal structure for a music festival, two beams represented by the lines:

$r: \begin{cases} x = 1 + 2\lambda \\ y = \lambda \\ z = 3 - \lambda \end{cases}$ and $s: \frac{x-2}{1} = \frac{y+1}{2} = \frac{z}{-1}$

must be placed.

An engineer needs to install a third reinforcing beam, $t$, that satisfies simultaneously:

1. It is perpendicular to the beams $r$ and $s$.

2. It passes through the point of intersection of the line $r$ with the plane $XY$.

Find the equation of the line $t$ in symmetric form and determine whether there is any point where this new beam $t$ intersects the beam $s$.

Given the lines $r$ and $s$ defined by:

$r: \frac{x-1}{a} = \frac{y}{1} = \frac{z+1}{2}$ and $s: \begin{cases} x + 2y - z = 3 \\ 2x - y + z = 1 \end{cases}$

1. a) Calculate the value of the parameter $a$ so that the lines $r$ and $s$ are perpendicular.
2. b) For the value of $a$ obtained, do the lines intersect or are they skew?
3. c) Find the equation of the plane that is parallel to $s$ and contains the line $r$.

Consider the plane $\pi: x + y + z = 2$ and the family of lines $r_m$ given by:

$r_m: \begin{cases} x - 2y + z = m \\ mx + y - z = 1 \end{cases}$

1. a) Find the value of $m$ so that the line $r_m$ is parallel to the plane $\pi$.
2. b) For that value of $m$, determine whether the line lies in the plane.
3. c) Calculate the equation of a plane $\pi'$ that is perpendicular to $\pi$ and contains the line $r_m$ when $m = 0$.

Find the equation of the line $t$ that perpendicularly intersects the lines $r$ and $s$, where:

$r: \frac{x}{1} = \frac{y-1}{-2} = \frac{z-3}{2}$ and $s: \begin{cases} x = 3 + 2\mu \\ y = 1 \\ z = \mu \end{cases}$

This exercise requires you to determine the direction of the "common perpendicular" and the auxiliary planes that allow you to locate the points of intersection on each line.

Given the points $A(1, 2, 3)$ and $B(3, 2, 1)$:

1. a) Determine the equation of the plane $\pi$ that is perpendicular to the segment $AB$ and passes through its midpoint (perpendicular bisector plane).
2. b) Given the line $r: \begin{cases} x - y = 0 \\ z = 4 \end{cases}$, study its relative position with respect to the plane $\pi$ found.
3. c) Find a point $C$ on the line $r$ such that the triangle $ABC$ is isosceles with non-equal side $AB$.

A telecommunications antenna is located at the point $A(5, 4, 3)$. A steel cable is to be installed perpendicular to a side wall represented by the plane $\pi: x + y = 10$.

1. a) Find the equation of the line $r$ containing the cable.
2. b) The cable must be anchored to a metal rail located on the wall, whose equation is the line $s: \begin{cases} x + y = 10 \\ x - z = 2 \end{cases}$. Check whether the cable (line $r$) and the rail (line $s$) intersect or are skew.
3. c) Calculate the minimum distance between the cable and the rail if they do not intersect.

In an exhibition hall in the shape of a rectangular cuboid with dimensions $10 \times 6 \times 4$ meters, a tensioned cable is installed connecting a vertex on the floor to the midpoint of the opposite top edge. A second cable connects the center of the ceiling to the midpoint of the vertical edge opposite that vertex.

1. a) Determine whether there is any point where the cables touch each other (intersecting lines) or if, on the contrary, they are skew in space without touching.
2. b) Does there exist any plane that is simultaneously parallel to both cables? Justify your answer based on the direction vectors of the cables.

In a geological stabilization project for a railway trench (ADIF), surveyors model the rock slope using the plane $\pi: 2x - 2y + z - 6 = 0$ (in meters). An anchor drilling machine advances along the directrix line $r: \frac{x-4}{1} = \frac{y-1}{2} = \frac{z-8}{-2}$.

1. a) Prove that the drilling does not hit the slope perpendicularly and find the coordinates of the impact point $P_0$ on the rock.
2. b) To install a row of monitoring markers in the rock, it is required to draw the orthogonal projection of the line $r$ onto the slope $\pi$. Reasonably obtain the general equation of the projecting plane of $r$ onto $\pi$ and write the equations of the projected trajectory as the intersection of two planes. Explain the physical meaning of this projected line in relation to the shadow that the drill rod would cast if illuminated by a distant light source oriented perpendicularly to the slope's surface.

The goal is to find an unknown plane $\pi$ that passes through the origin of coordinates $O(0, 0, 0)$ and simultaneously satisfies two geometric conditions:

• Condition 1: The plane $\pi$ is perpendicular to the plane $\pi_1: x - y + 2z - 1 = 0$.

• Condition 2: The orthogonal projection of the line $r: \frac{x-1}{1} = \frac{y}{1} = \frac{z+2}{0}$ onto the plane $\pi$ reduces to a single point.

1. a) Using exclusively Condition 2, deduce the direction of the normal vector $\vec{n}_\pi$ of the desired plane in a reasoned manner, without calculating projecting planes.
2. b) Check whether Condition 1 is compatible with the direction deduced in part a). If it is, write the general equation of the plane $\pi$; if not, explain with reasoning what geometric impossibility would arise.
3. c) Obtain the continuous equations of a line $s$ that passes through the point $A(2, 3, -1)$ and is parallel to both planes, $\pi$ and $\pi_1$, at the same time. Justify why the direction vector of $s$ must coincide with the direction of the line of intersection between $\pi$ and $\pi_1$.