Two delivery drones fly in straight lines over Mexico City. Their paths are not parallel, but one passes above the other and they never collide. In the plane this is impossible: two lines either intersect or are parallel. In space there is a third option. On this page you will learn to classify every case between lines and planes with a single method: comparing vectors and computing ranks.
A line in space: through two points and whether a point lies on it
A line is determined by a point $A$ and a direction vector $\vec u$. Its parametric equations are $X=A+t\,\vec u$, that is, $x=a_1+tu_1$, $y=a_2+tu_2$, $z=a_3+tu_3$.
Practice★☆☆Not started
An observation satellite follows a linear trajectory in space defined by line $r: (x, y, z) = (1, 2, 0) + \lambda(2, -1, 3)$. It is desired to check whether the…
An observation satellite follows a linear trajectory in space defined by line $r: (x, y, z) = (1, 2, 0) + \lambda(2, -1, 3)$. It is desired to check whether the satellite will pass exactly through a control point located at $P(5, 0, 6)$. Determine whether point $P$ belongs to the satellite's trajectory.
Two lines: coincident, parallel, intersecting, or skew
The common perpendicular of two skew lines
Two skew lines do not touch, but there is a unique line that intersects both and is perpendicular to both. It marks the shortest distance between them.
Back to the beginning: the two drones have different directions and $\vec{AB}$ is not in their plane, so their paths are skew and there is no risk of collision.
Practice★☆☆Not started
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…
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…
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 $B$?
Two aerial surveillance drones follow straight trajectories defined by lines $r$ and $s$. The control center needs to know if there is any risk of collision (if the…
Two aerial surveillance drones follow straight trajectories defined by lines $r$ and $s$. The control center needs to know if there is any risk of collision (if the trajectories intersect). The equations of the trajectories are:
$r: \frac{x-1}{2} = \frac{y}{1} = \frac{z+2}{3}$
$s: \begin{cases} x + y - z = 0 \\ 2x - y + 3z = 1 \end{cases}$
Determine the relative position of both trajectories.
Given the lines $r$ and $s$ with equations: $r: \frac{x-2}{1} = \frac{y+1}{2} = \frac{z}{-1}$ $s: \begin{cases} x - y + 2z = 4 \\ 2x + z = 1 \end{cases}$ a) Show…
Given the lines $r$ and $s$ with equations:
$r: \frac{x-2}{1} = \frac{y+1}{2} = \frac{z}{-1}$
$s: \begin{cases} x - y + 2z = 4 \\ 2x + z = 1 \end{cases}$
a) Show analytically that lines $r$ and $s$ are skew.
b) Find the equation of the common perpendicular to both lines.
c) Determine the relative position of this common perpendicular with the coordinate plane $XZ$.
Line and plane: contained, parallel, or intersecting
If the line is given as the intersection of two planes, its direction is $\vec v=\vec n_1\times\vec n_2$ (cross product of their normals). You can also solve the system and leave one free parameter.
Practice★☆☆Not started
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…
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…
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.
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 $\pi: x + y + z - 1 = 0$.…
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.
Consider plane $\pi: 2x - y + az + 3 = 0$ and line $r$ passing through points $A(1, 0, -1)$ and $B(2, 1, 0)$. Determine the value of parameter $a \in \mathbb{R}$ so that…
Consider plane $\pi: 2x - y + az + 3 = 0$ and line $r$ passing through points $A(1, 0, -1)$ and $B(2, 1, 0)$. Determine the value of parameter $a \in \mathbb{R}$ so that the line is parallel to the plane. For this value of $a$, is the line contained in the plane? Justify your answer.
In the design of a smart building, two flat surfaces corresponding to the ceiling and floor of a mechanical room have been defined. The equations of the planes are…
In the design of a smart building, two flat surfaces corresponding to the ceiling and floor of a mechanical room have been defined. The equations of the planes are $\pi_1: 2x - y + 3z - 5 = 0$ and $\pi_2: 4x - 2y + 6z + 8 = 0$. Analyze the relative position of both planes to confirm whether they are parallel.
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…
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.
Three planes form a system of three equations with three unknowns. Let $M$ be the coefficient matrix and $M^*$ the augmented matrix (with the column of constant terms). This is the same analysis of linear systems from earlier semesters, now with geometric meaning.
Practice★★☆Not started
Discuss, as a function of the real parameter $m$, the relative position of the following three planes: $\pi_1: mx + y + z = 1$ $\pi_2: x + my + z = m$…
Discuss, as a function of the real parameter $m$, the relative position of the following three planes:
In an industrial design project, three surfaces are defined by the following planes in space: $\pi_1: x + ay + z = a$ $\pi_2: ax + y + (a-1)z = 1$ $\pi_3: x + y + z = 1$…
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.
Constructing planes: through three points, parallel to a line, and a pencil of planes
Many problems ask for a plane that satisfies a condition with respect to points or lines. There are three tools.
Practice★★☆Not started
A pencil of planes has line $r: \begin{cases} 3x - y + z = 2 \\ x + 2y - z = 1 \end{cases}$ as its axis. Find which plane in the pencil passes through point…
A pencil of planes has line $r: \begin{cases} 3x - y + z = 2 \\ x + 2y - z = 1 \end{cases}$ as its axis. Find which plane in the pencil passes through point $P(1, 1, 1)$. Once obtained, determine the relative position of this plane with the $OY$ axis.
Given lines $r$ and $s$, show that they are skew and find the equation of the plane containing $s$ and parallel to $r$: $r: \frac{x}{1} = \frac{y-1}{2} = \frac{z+1}{1}$…
Given lines $r$ and $s$, show that they are skew and find the equation of the plane containing $s$ and parallel to $r$:
Consider a drone trajectory represented by the line $r$ passing through point $A(1, 2, -1)$ and having direction vector $\vec{v} = (k, 1, 2)$. On the other hand, a solar…
Consider a drone trajectory represented by the line $r$ passing through point $A(1, 2, -1)$ and having direction vector $\vec{v} = (k, 1, 2)$. On the other hand, a solar panel is contained in plane $\pi$ defined by points $P(1, 0, 1)$, $Q(0, 2, 3)$ and $R(2, 1, 0)$.
Determine the value of the parameter $k$ so that the drone's trajectory is parallel to the solar panel. For this value of $k$, is there any line $s$ contained in plane $\pi$ that is perpendicular to $r$? Justify your answer based on the relative position of the elements.