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.
NoteParallel 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.
