Exercises: Equation of a plane in space

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.

The surface of a solar panel on a satellite can be represented by the following parametric equations:

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

Obtain the general or implicit equation of this plane.

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.

Given the equation of the plane representing one of the walls of a room, $\pi \equiv 2x - 3y + z - 6 = 0$, determine, giving reasons, whether the following points belong to this wall:

1. a) $Q(3, 0, 0)$
2. b) $R(1, 1, 1)$

Find the equation of the plane passing through point $A(5, -2, 4)$ and parallel to the plane with equation $\pi \equiv 3x - y + 2z + 8 = 0$. Remember that two parallel planes share the same normal vector.

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 $Ax + By + Cz + D = 0$.

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 $M(0, 4, -2)$.

Determine the equation of the plane containing the line $r \equiv \begin{cases} x = 2 - t \\ y = t \\ z = 1 + 2t \end{cases}$ and also passing through the external point $P(1, 1, 1)$.

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 meters.

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).

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.

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 $B(3, 0, 5)$.

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.

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.

In a rescue mission, a drone moves along a straight trajectory given by the line $r \equiv \frac{x-1}{2} = \frac{y}{3} = \frac{z+2}{-1}$. The drone must emit a radio frequency signal that is only effective if it hits a flat safety screen perpendicularly.

1. a) Determine the equation of the plane representing this screen if it must pass through point $P(2, 2, 2)$ and be perpendicular to the drone's trajectory.
2. b) Calculate the exact point where the drone's trajectory would hit the screen.

Given the planes $\pi_1 \equiv (a-2)x + 2y + z = 0$ and $\pi_2 \equiv 4x + ay + 2z = b$:

1. a) Calculate the values of $a$ and $b$ so that the planes are coincident.
2. b) Calculate the values of $a$ and $b$ so that the planes are parallel but not coincident.
3. c) What relationship must exist between $a$ and $b$ so that the origin of coordinates belongs to the intersection of both planes?

Find the equation of the plane containing the following lines, first verifying that they intersect at a point:

If a third laser beam follows the direction of the normal vector to this plane and originates from the point of intersection of $r$ and $s$, what would its vector equation be?

An architect is designing the roof of a sports pavilion. The main structure is defined by two metal beams located along the lines:

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

Determine the general equation of the plane $\pi$ that is parallel to both beams and is at a distance of $\sqrt{14}$ units from the origin. Is there a unique solution? Reason your answer.

Consider the pencil of planes whose axis is the line $r: \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z}{3}$. Calculate the equation of the plane $\pi$ belonging to this pencil that satisfies the following two conditions simultaneously:

1. It is perpendicular to the plane $\sigma: x - 2y + z - 5 = 0$.

2. The point $P(0, a, 1)$ belongs to the plane $\pi$ for some real value of $a$.

Determine the value of $a$ and the general equation of $\pi$.

A laser beam starts from the point $A(1, 2, -2)$ and travels in the direction of the vector $\vec{u} = (1, -1, 1)$ until it strikes a plane mirror containing the line $s: \begin{cases} x - y + z = 2 \\ 2x + y - z = 1 \end{cases}$ and the point $B(0, 0, 4)$.

1. a) Calculate the equation of the plane $\pi$ where the mirror is located.
2. b) Find the coordinates of the point of impact $P$ of the laser on the mirror.

Find the equation of the plane $\pi$ that contains the line of intersection of the planes $\alpha: 2x - y + 3z = 1$ and $\beta: x + 2y - z = 0$, and that intersects the positive coordinate semi-axes $OX$, $OY$, and $OZ$ at points $P$, $Q$, and $R$ such that the volume of the tetrahedron formed by the origin and these points is minimized.

Consider the intersecting lines $r$ and $s$ defining a plane $\alpha$. Line $r$ passes through $P(1, 0, 1)$ with direction $\vec{d_r} = (2, 1, -1)$ and line $s$ passes through $Q(1, 0, 1)$ with direction $\vec{d_s} = (0, 1, 1)$.

Calculate the equation of a plane $\pi$ parallel to $\alpha$ such that the sphere with center $C(3, 2, 1)$ and radius $R=2$ is tangent to this plane $\pi$. Obtain the two possible solutions.

Given the points $A(1, 1, 1)$, $B(3, 0, 2)$ and $C(4, k, 0)$, study the relative position of the planes $\pi_1$, $\pi_2$ and $\pi_3$ depending on the real parameter $k$, knowing that:

• $\pi_1$ is the plane that passes through $A$, $B$ and the origin.

• $\pi_2$ is the plane that contains $C$ and is perpendicular to the line passing through $A$ and $B$.

• $\pi_3$ is the plane with equation $x + y + z = k + 2$.

A drone moves in a straight line following the path of line $r: \frac{x-1}{2} = \frac{y-1}{1} = \frac{z-1}{2}$. It is desired to install a flat protective screen $\pi$ containing line $s: \begin{cases} x = t \\ y = 2 \\ z = 1-t \end{cases}$.

Find the equation of plane $\pi$ such that it is parallel to the drone's path. Calculate the minimum distance at which the drone will pass from this screen.

Let $P(1, 2, -1)$ be a point in space and $r$ be the line of intersection of the planes $x - y = 0$ and $z = 2$.

1. a) Find the equation of the plane $\pi_1$ containing $r$ and passing through $P$.
2. b) Find the equation of the plane $\pi_2$ containing $r$ and perpendicular to $\pi_1$.
3. c) Calculate the angle formed by the plane $\pi_2$ with the coordinate plane $XY$.