Relative positions

Given the lines $r$ and $s$ expressed by their parametric equations:

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

Study the relative position of both by comparing their direction vectors. Are they parallel or do they have different directions?

Given the points $A(1, 0, -1)$, $B(2, 1, 0)$ and $C(3, 2, 1)$, determine the relative position of the line passing through $A$ and $B$ with respect to the line passing through $B$ and $C$. Is it the same line (coincident) or are they intersecting lines at the point $B$?

Study the relative position of the following lines $r$ and $s$. If they intersect, calculate the coordinates of the point of intersection:

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

$s: \frac{x-2}{2} = \frac{y+1}{1} = \frac{z-4}{-1}$

A laser beam propagates along the line $r: \frac{x-1}{2} = \frac{y+2}{1} = \frac{z}{3}$ and hits a flat sheet of glass whose equation is $\pi: x + y + z - 1 = 0$. Determine the relative position between the laser beam and the sheet. If they intersect, calculate the coordinates of the impact point.

Check whether the line $r$, defined as the intersection of the planes $x + y = 2$ and $x - y = 0$, is parallel to the coordinate plane $XY$ (the $z = 0$ plane). Justify your answer based on the direction vector of the line.

An architect uses design software to place a structural beam represented by the line $r: \dfrac{x-2}{3} = \dfrac{y+1}{-2} = \dfrac{z-4}{4}$. Determine whether the beam is perpendicular to the facade plane $\pi: 3x - 2y + 4z + 1 = 0$. Justify your answer.

Determine the value of the parameter $k$ so that the planes $\pi_1: x + ky + z - 2 = 0$ and $\pi_2: 2x - 4y + 2z + 1 = 0$ are parallel. For that value of $k$, could the planes be coincident? Justify your answer by observing the constant terms.

Discuss, as a function of the real parameter $m$, the relative position of the following three planes:

$\pi_1: mx + y + z = 1$

$\pi_2: x + my + z = m$

$\pi_3: x + y + mz = m^2$

In an industrial design project, three surfaces are defined by the following planes in space:

$\pi_1: x + ay + z = a$

$\pi_2: ax + y + (a-1)z = 1$

$\pi_3: x + y + z = 1$

Discuss the relative position of the three planes according to the different values of the real parameter $a$. In cases where the three planes intersect in a line (forming a pencil of planes), obtain the parametric equation of this line.