Exercises: Distances in space

In a 3D video game development environment, two objects are located at points $A(2, -3, 1)$ and $B(6, 1, 1)$, where units are in metres. Calculate the distance separating both objects to program a collision if it is less than 5 metres.

A mobile phone transmitter antenna is located at point $P(1, 2, -3)$. It is known that the signal is optimally intercepted by a flat reflecting panel whose equation is $\pi: 3x - 4z + 2 = 0$. Determine the minimum distance the wave must travel from the antenna to the panel.

In the design of an exhibition hall, the floor and ceiling have been modelled as two parallel planes with equations $\pi_1: x - 2y + 2z - 4 = 0$ and $\pi_2: x - 2y + 2z + 11 = 0$. Find the height of the hall.

The hook of a construction crane is suspended at position $Q(4, -3, 12)$. The technicians need to know the distance from the hook to the z-axis ($OZ$-axis). Calculate this distance to verify the crane's turning clearance relative to the vertical reference axis.

Consider the point $A(1, 5, -2)$ and the line $r$ defined by its parametric equations:

Find the distance from point $A$ to line $r$ using the vector product formula.

A laser depth sensor is located at point $S(5, -7, 4)$. The laser points directly towards the side wall, represented by the coordinate plane $XZ$. How far is the sensor from this wall?

A metal rail has been installed for an industrial robot following the trajectory of the line:

If the factory floor is the plane $\pi: x - 2z + 5 = 0$, first check that the rail is parallel to the floor and then calculate the safety distance of the rail from the floor.

Calculate the coordinates of the points on the x-axis (axis $OX$) whose distance to the point $P(3, 2, 2)$ is exactly $\sqrt{8}$ units. Explain geometrically why you obtain a unique solution.

In a 2nd year of Bachillerato exam, students are asked to calculate the distance between the two parallel planes $\pi_1: 2x - 2y + z - 6 = 0$ and $\pi_2: 4x - 4y + 2z + 8 = 0$. Lucas writes the following solution on his exam:

"Since the planes are parallel, I directly subtract the constant terms and divide by the magnitude of the normal vector of $\pi_1$:

1. Identify the conceptual error Lucas makes when applying the formula directly with the given coefficients.

2. Explain geometrically why the simplified formula of subtracting constant terms is only valid when the equations have exactly the same normal vector (or how it would be safely solved by choosing a point on one of the planes).

3. Correct the calculation and find the real, exact distance between $\pi_1$ and $\pi_2$.

Consider the point $P(4, 3, 7)$ and the vertical line $r \equiv \begin{cases} x = 1 \\ y = -1 \end{cases}$, which is parallel to the $Z$-axis.

1. Estimate and reason: Before applying any formula, analyze the position of $P$ and of the line $r$. Explain why the height ($z$-coordinate) of $P$ should not affect the minimum distance to the vertical line $r$, and deduce the exact value that this distance should have based solely on the $x$ and $y$ coordinates projected onto the plane $z = 0$.

2. Check: Write the line $r$ in continuous or parametric form, choose a point $A \in r$, its direction vector $\vec{v}$, and formally calculate the distance applying the three-dimensional cross product formula: $d(P, r) = \frac{|\vec{AP} \times \vec{v}|}{|\vec{v}|}$. Check whether the result matches your previous deduction and reason what geometric magnitude the vector $\vec{AP} \times \vec{v}$ represents in this context.

A sounding rocket follows a linear trajectory defined by the line $r \equiv \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 + 2\lambda \end{cases}$. The rocket must send a signal to the tracking station located at point $P(5, -1, 4)$. Calculate the minimum distance at which the rocket will pass the tracking station during its flight.

The support structures of a pavilion are being designed. Two of the main beams are modelled by the lines $r \equiv \frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{-1}$ and $s \equiv \frac{x}{1} = \frac{y-2}{2} = \frac{z}{3}$. Determine the minimum distance separating both beams to check whether it is necessary to install a safety connector between them.

In an archaeological excavation, it has been determined that the original floor of a room lies on the plane $\pi \equiv 2x - y + 2z - 6 = 0$. A set of sensors moves along a metal track represented by the line $r \equiv \begin{cases} x = \mu \\ y = 4\mu - 1 \\ z = \mu + 1 \end{cases}$.

1. a) Verify that the track is parallel to the room's floor.
2. b) Calculate the constant distance between the sensors and the floor.

Find the coordinates of the points on the line $r \equiv x = y = z$ that are at a distance of $2\sqrt{3}$ units from the plane $\pi \equiv x + 2y - 2z + 1 = 0$.

Determine the locus of points in space that are equidistant from points $A(1, -2, 3)$ and $B(3, 0, 1)$. What geometric figure does it represent? Calculate the distance from the origin of coordinates to this locus.

Consider the cube with edge length 4 units, having one vertex at the origin $(0,0,0)$ and its edges lying along the positive coordinate axes. Calculate the distance from vertex $V(4, 4, 4)$ to the cube's diagonal joining points $(4, 0, 0)$ and $(0, 4, 0)$.

A laser light beam leaves point $A(1, 1, 1)$ in the direction of vector $\vec{v} = (1, 0, 2)$. We want to place a flat mirror parallel to plane $XY$ such that the distance from point $A$ to the mirror is 5 units.

1. a) Find the two possible equations of the planes containing the mirror.
2. b) Calculate the distance from the origin to each of those planes.

Calculate the value of the parameter $k$ so that the distance between the parallel planes $\pi_1 \equiv 2x + 2y - z + 3 = 0$ and $\pi_2 \equiv 4x + 4y - 2z + k = 0$ is 3 units. Explain why there are two possible values for $k$.

An underground ventilation tunnel is to be designed, whose central axis follows the line:

To protect it, two flat screens will be built parallel to the family of planes with normal vector $\vec{n} = (2, 1, -2)$.

1. Explain with reasoning why, if the line $r$ were not parallel to the planes of this family, it would make no sense to search for planes in the family that are at a constant non-zero distance from the line. Check analytically that in this case $r$ is indeed parallel to these planes.

2. Write the general equation of the family of parallel planes $\pi_D \equiv 2x + y - 2z + D = 0$ and determine the two possible values of the parameter $D$ such that the distance from the line $r$ to the corresponding plane is exactly $5\text{ metres}$.

3. From the values of $D$ found, deduce directly (without applying the distance formula again) the distance separating both flat screens from each other, justifying your answer.

Spain's air navigation service provider (ENAIRE) monitors in its control center the trajectory of two commercial aircraft in route within peninsular airspace. Taking as reference a Cartesian coordinate system with distances in kilometers, the straight-line trajectories of both flights are described by:

1. Show analytically that the trajectories are neither parallel nor intersect at any point (that is, that they are skew lines in space).

2. The air safety protocol requires issuing a preventive alert if the theoretical minimum distance between the linear trajectories is less than $8\text{ km}$. Calculate the minimum distance between both lines using the scalar triple product and cross product formula, and determine whether the control center must issue such an alert.

3. From the perspective of the actual movement of the aircraft, explain why a distance between trajectories of less than $8\text{ km}$ does not necessarily imply an imminent risk of physical collision, indicating what additional kinematic quantity should be considered to know whether both aircraft coincide near the point of closest approach.

Two communications satellites follow linear paths in space (measured in hundreds of kilometres) represented by the lines $r$ and $s$:

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

Find:

1. Determine the values of the parameter $k$ so that the minimum distance between the two paths is exactly $\sqrt{6}$ units (600 km).

2. For each value of $k$ obtained in the previous section, calculate the points on each path where this minimum distance is reached (points of the common perpendicular).

A logistics company uses drones to transport packages. A drone travels from point $A(1, 0, 2)$ in a straight line towards point $B(5, 4, 10)$. In the centre of the operating area, there is a transmitting antenna located at point $P(3, 5, 1)$.

Calculate:

1. The exact point on the drone's path that is at the minimum distance from the antenna.

2. If the signal from the antenna has a range of $5$ units, determine during which interval of its route (values of the parameter $\lambda$) the drone has coverage.

Consider a plane $\pi$ passing through the point $M(1, 0, 3)$ and perpendicular to the line $r: \frac{x-2}{1} = \frac{y+1}{2} = \frac{z}{1}$.

1. Find the equation of the sphere tangent to the plane $\pi$ with its centre at the point $C(4, 5, 2)$.

2. Calculate the distance from the origin of coordinates to the point of tangency between the sphere and the plane.

An architect designs an avant-garde structure made up of two tensioned cables. In a Cartesian coordinate system, the first cable coincides with the line $r$ passing through $A(1, 1, 1)$ and $B(2, 3, 0)$. The second cable follows the direction of vector $\vec{u} = (1, 0, 1)$ and passes through point $C(0, 2, 2)$.

Calculate:

1. The minimum distance between the two cables.

2. The equation of the line representing a metallic reinforcement joining both cables along the shortest possible path (common perpendicular).

Find the coordinates of the points on the line $r: \frac{x-1}{2} = \frac{y+2}{3} = \frac{z-3}{-1}$ that are equidistant from the planes:

$\pi_1: x + 2y - 2z + 1 = 0$

$\pi_2: 8x - 4y + z - 5 = 0$

Analyse whether there is any point on the line whose distance to both planes is zero and explain its geometric meaning.

Consider the point $P(1, 2, -1)$ and the line $r: \begin{cases} x + y = 3 \\ 2x - z = 4 \end{cases}$.

1. Calculate the symmetric point of $P$ with respect to the line $r$. Call it $P'$.

2. Determine the distance from $P'$ to the plane $\pi: 2x - 3y + z + 10 = 0$.

3. What is the volume of the tetrahedron formed by the point $P$, its projection onto the line $r$, and the points of intersection of the plane $\pi$ with the axes $OX$ and $OY$?

A laser beam starts from point $A(5, 2, 6)$ and reflects off a flat mirror represented by the coordinate plane $XY$ ($z=0$). The reflected beam must pass through point $B(1, 4, 2)$.

Find:

1. The coordinates of the impact point $C$ of the laser on the mirror.

2. The total distance travelled by the beam from point $A$ to point $B$. (Note: The minimum distance is obtained using the symmetric point).

Given the family of planes $\pi_a: ax + (a-1)y + z + (a-2) = 0$, where $a \in \mathbb{R}$:

1. Show that all planes in the family contain the same common line $r$. Find the equations of this line.

2. Calculate the distance from point $Q(2, -1, 0)$ to the line $r$.

3. Determine the value of $a$ for which the plane $\pi_a$ is at a distance of $\frac{1}{\sqrt{2}}$ units from point $P(0, 0, 0)$.

Given the point $P(2, -1, 3)$ and the line $r \equiv \frac{x-1}{2} = \frac{y-2}{1} = \frac{z+1}{-2}$. An exercise proposes solving two successive questions:

(a) Calculate the distance from point $P$ to line $r$.

(b) Find the coordinates of point $P'$, the symmetric of $P$ with respect to $r$.

Two students discuss the solution strategy:

• Lucía proposes solving (a) using the cross product formula $d(P, r) = \frac{|\vec{AP} \times \vec{v}|}{|\vec{v}|}$, and then solving (b) by determining the orthogonal projection $M$ of $P$ onto $r$.

• Carlos argues that using the cross product formula in this exercise is a poor methodological choice that generates redundant calculations, and that it is more efficient to start directly by finding the orthogonal projection $M$.

You are asked to:

1. Critically evaluate both positions: explain why Lucía's method duplicates the calculation effort and show how obtaining the orthogonal projection $M$ first allows you to answer parts (a) and (b) immediately.

2. Following the method proposed by Carlos (using a generic point of the line or constructing the auxiliary perpendicular plane to $r$ passing through $P$), find the coordinates of the orthogonal projection $M$, deduce from it the exact distance $d(P, r)$, and calculate the coordinates of the symmetric point $P'$.

Two intersecting planes $\pi_1 \equiv 2x - y + 2z - 1 = 0$ and $\pi_2 \equiv x + 2y - 2z + 5 = 0$ intersect along a line in space, dividing it into four dihedral angular regions.

1. Using the analytical definition of the distance from a point to a plane, set up the geometric condition that any point $X(x, y, z)$ in space must satisfy to be equidistant from both planes: $d(X, \pi_1) = d(X, \pi_2)$.

2. Expand the resulting equation applying the properties of absolute value and obtain the general equations of the two resulting bisector planes, $\sigma_1$ and $\sigma_2$.

3. From the normal vectors of $\sigma_1$ and $\sigma_2$, verify algebraically that both bisector planes are perpendicular to each other.

4. Generalize the previous result for any pair of intersecting planes: if two planes have unit normal vectors $\vec{u}$ and $\vec{v}$ (with $\vec{u} \neq \pm \vec{v}$), prove using the properties of the dot product that the normal direction vectors of the bisector planes, $\vec{n}_1 = \vec{u} + \vec{v}$ and $\vec{n}_2 = \vec{u} - \vec{v}$, are always orthogonal, explaining the geometric meaning of this fact.