Equation of a plane in space

Determine the equation of the plane $\pi$ containing the line $r$ given by:

and parallel to line $s$, which passes through points $P(0, 2, 1)$ and $Q(1, 2, 4)$. Justify why the direction vector of $s$ cannot be proportional to that of $r$ in order for the plane to be uniquely determined.

Find the value of the parameter $k \in \mathbb{R}$ so that the points $A(1, 0, 1)$, $B(0, 2, 1)$, $C(2, 1, 0)$ and $D(k, k, 2)$ are coplanar. Once $k$ is found, write the equation of the plane containing them in parametric form.

An engineer is designing the base of a component using CAD software. The base must be a plane passing through point $P(2, -1, 3)$ with a normal vector $\vec{n} = (3, 2, -4)$. Calculate the general equation of the plane that defines this base.

It is desired to install a spotlight whose beam path is perpendicular to a metal plate. If the path of the light beam is given by the line $r \equiv \frac{x-1}{2} = \frac{y+3}{-1} = \frac{z}{4}$, find the equation of the plane containing the metal plate, knowing that it passes through the point $M(0, 4, -2)$.

In a virtual reality scenario, three sensors are located at points $A(1, 0, 0)$, $B(0, 2, 0)$ and $C(0, 0, 3)$. Determine the general equation of the plane containing these three sensors.

An architect is designing a flat roof for a small pavilion. The roof must rest on three pillars whose upper ends are located at the points $A(2, 0, 3)$, $B(0, 4, 3)$ and $C(1, 1, 5)$, where the units are in metres.

1. a) Obtain the general or implicit equation of the plane containing the roof.
2. b) Calculate the points of intersection of this plane with the coordinate axes to determine where the anchors would be located if the roof extended to touch the main walls (axes).

An architect uses the "intercept equation" to define a sloping roof that intersects the coordinate axes at points $(4, 0, 0)$, $(0, 3, 0)$ and $(0, 0, 2)$. Write its equation directly and convert it to the general form $Ax + By + Cz + D = 0$.

A plane $\pi$ intersects the coordinate axes at points $X(a, 0, 0)$, $Y(0, b, 0)$ and $Z(0, 0, c)$. It is known that the plane passes through the point $P(1, 2, 2)$ and that the distances from the origin to the intersection points are equal ($a = b = c$).

1. a) Write the general equation of the plane.
2. b) Calculate the volume of the tetrahedron formed by the origin and the three intersection points with the axes.

Consider the pencil of planes whose axis is the line of intersection of the planes $\pi_1 \equiv 2x - y + z - 3 = 0$ and $\pi_2 \equiv x + y - 2z + 1 = 0$. Find the plane belonging to this pencil that passes through the midpoint of the line segment with endpoints $A(1, 2, -1)$ and $B(3, 0, 5)$.