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 meters. Calculate the distance separating both objects to program a collision if it is less than 5 meters.

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.

A cell 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.

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?

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 $\pi \equiv x + 2y - 2z + 1 = 0$.

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 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.

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 $(0, 4, 0)$.

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.

Two communications satellites follow linear paths in space (measured in hundreds of kilometers) 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).

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).

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.

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.

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 $k$.

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.

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.

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 center 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.

A logistics company uses drones to transport packages. A drone travels from point $A(1, 0, 2)$ in a straight line toward point $B(5, 4, 10)$. In the center 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.

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

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 $OY$?

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 $P(0, 0, 0)$.