Classification of Lines in Three-Dimensional Space
Types of Line Relationships
In three-dimensional space, lines can be classified into four main categories based on their relative positions:
- Coincident Lines
- Parallel Lines
- Intersecting Lines
- Skew Lines
Coincident Lines
- Coincident lines are lines that occupy the same position in space. In other words, they are the same line.
Consider the lines: $L_1: (x, y, z) = (1, 2, 3) + t(1, 1, 1)$ $L_2: (x, y, z) = (1, 2, 3) + s(2, 2, 2)$
These lines are coincident because $L_2$ can be obtained from $L_1$ by substituting $t = 2s$.
Parallel Lines
- Parallel lines are lines that maintain a constant distance between them and never intersect, even when extended infinitely.
The lines: $L_1: (x, y, z) = (1, 2, 3) + t(1, 1, 1)$ $L_2: (x, y, z) = (4, 5, 6) + s(1, 1, 1)$
are parallel because they have the same direction vector $(1, 1, 1)$ but different points of origin.
Parallel lines in 3D space have direction vectors that are scalar multiples of each other.
Intersecting Lines
- Intersecting lines are lines that cross each other at a single point in space.
Consider the lines: $L_1: (x, y, z) = (1, 0, 0) + t(1, 1, 0)$ $L_2: (x, y, z) = (0, 1, 0) + s(1, -1, 0)$
These lines intersect at the point $(1, 1, 0)$ when $t = 1$ and $s = 1$.
Skew Lines
- Skew lines are non-parallel lines that do not intersect in three-dimensional space.
- This is a unique property of 3D geometry that doesn't exist in 2D.
The lines:
$$\begin{aligned}
& L_1:(x, y, z)=(1,0,0)+t(1,1,0) \\
& L_2:(x, y, z)=(0,0,1)+s(0,1,1)
\end{aligned}$$
are potentially skew. To confirm, we check whether there exist values of $s$ and $t$ such that the parametric equations are equal:
$$\begin{gathered}
1+t=0 \quad \text { (for } x \text { coordinates) } \\
0+t=0+s \quad \text { (for } y \text { coordinates) } \\
0=1+s \quad \text { (for } z \text { coordinates) }
\end{gathered}$$
Solving these equations:
1. From $1+t=0$, we get $t=-1$.
2. From $t=s$, substituting $t=-1$, we get $s=-1$.
3. From $0=1+s$, we solve for $s$ and get $s=-1$.
However, these values contradict the second equation, where $t=s$ should hold. Since no single pair $(s, t)$ satisfies all three equations simultaneously, the lines do not intersect.
Since they also have different direction vectors, they are skew lines, meaning they neither intersect nor are parallel.
Skew lines can be thought of as lines that "pass by" each other in 3D space without touching.
Finding Points of Intersection
- When lines intersect, it's often necessary to find the point of intersection.
- This can be done by equating the parametric equations of the lines and solving for the parameters.
For the intersecting lines: \[ L_1 : (x, y, z) = (1, 0, 0) + t(1, 1, 0) \]\[ L_2 : (x, y, z) = (0, 1, 0) + s(1, -1, 0) \] Equating the components, we form the system of equations: \[\begin{cases} 1 + t = s \\ t = 1 - s \\ 0 = 0 \end{cases} \] Solving these equations, we obtain \( t = 1 \) and \( s = 1 \), which when substituted back into either line equation gives the point of intersection: \[ (2, 1, 0). \]
Students often forget that in 3D, non-parallel lines don't necessarily intersect. Always check if the lines are skew before attempting to find an intersection point.
Connection to Systems of Linear Equations
- The classification of lines and finding points of intersection are closely linked to solving systems of linear equations (AHL 1.16).
- When finding intersection points, we essentially solve a system of linear equations.
For the lines: \[ L_1 : (x, y, z) = (1, 2, 3) + t(1, 1, 1) \]\[ L_2 : (x, y, z) = (2, 3, 4) + s(2, 2, 2) \] To find if they intersect, we solve the system: \[\begin{cases} 1 + t = 2 + 2s \\ 2 + t = 3 + 2s \\ 3 + t = 4 + 2s \end{cases} \] This system has no solution, indicating that the lines are skew.
When solving systems of equations for line intersections, if you get a consistent system with one free variable, the lines are coincident. If you get an inconsistent system, the lines are either parallel or skew.
Geometric Interpretation
Understanding the classification of lines in 3D space is crucial for visualizing and solving geometric problems.
- Coincident lines represent the same path in space.
- Parallel lines maintain a constant distance and never meet.
- Intersecting lines share a common point.
- Skew lines pass by each other without intersecting or being parallel.
The ability to classify lines and find intersections is fundamental in many areas of mathematics and its applications, including computer graphics, robotics, and physics.
