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.

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 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.

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

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 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.

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).

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.

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 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 $k$.