Mathwords logoMathwords

Skew Lines — Definition, Examples & Properties

Skew Lines

Lines in three dimensional space that do not intersect and are not parallel.

 

Two arrows crossing at a point, angled in different directions to suggest lines in 3D space that neither intersect nor run...

Worked Example

Problem: Determine whether the following two lines in 3D space are skew: Line 1 passes through point (1, 0, 0) with direction vector ⟨0, 1, 0⟩, and Line 2 passes through point (0, 0, 1) with direction vector ⟨1, 0, 0⟩.
Step 1: Write the parametric equations for each line. Line 1 uses parameter t and Line 2 uses parameter s.
Line 1: (x,y,z)=(1,0,0)+t⟨0,1,0⟩=(1,  t,  0)\text{Line 1: } (x, y, z) = (1, 0, 0) + t\langle 0, 1, 0 \rangle = (1,\; t,\; 0)
Step 2: Write the parametric equations for Line 2.
Line 2: (x,y,z)=(0,0,1)+s⟨1,0,0⟩=(s,  0,  1)\text{Line 2: } (x, y, z) = (0, 0, 1) + s\langle 1, 0, 0 \rangle = (s,\; 0,\; 1)
Step 3: Check if the lines are parallel by comparing direction vectors. Two lines are parallel if one direction vector is a scalar multiple of the other. Here, ⟨0, 1, 0⟩ is not a scalar multiple of ⟨1, 0, 0⟩, so the lines are NOT parallel.
⟨0,1,0⟩≠k⟨1,0,0⟩ for any scalar k\langle 0, 1, 0 \rangle \neq k\langle 1, 0, 0 \rangle \text{ for any scalar } k
Step 4: Check if the lines intersect by setting their coordinates equal and solving. From the x-coordinates: 1 = s. From the y-coordinates: t = 0. From the z-coordinates: 0 = 1.
x:  1=s,y:  t=0,z:  0=1  (contradiction!)x:\; 1 = s, \quad y:\; t = 0, \quad z:\; 0 = 1 \; \text{(contradiction!)}
Step 5: The z-equation gives 0 = 1, which is impossible. This means no values of t and s satisfy all three equations simultaneously, so the lines do not intersect.
Answer: The lines are not parallel and do not intersect, so they are skew lines.

Another Example

Problem: Think of a real-world example of skew lines.
Step 1: Consider a bridge that passes over a highway. The road on the bridge runs east-west, while the highway below runs north-south. These two roads are at different heights (different planes) and point in different directions.
Step 2: The bridge road and the highway never touch each other, and they are not parallel because they run in perpendicular directions. Since they exist at different elevations and never cross, they satisfy the definition of skew lines.
Answer: The road on a bridge and the highway passing beneath it (running in a different direction) are a real-world model of skew lines.

Frequently Asked Questions

Can skew lines exist in 2D?
No. In two dimensions, any two lines that are not parallel must eventually intersect. Skew lines require three-dimensional space because the lines need to be offset in a direction that is not available on a flat plane.
How do you tell the difference between skew lines and parallel lines?
Parallel lines have the same direction (their direction vectors are scalar multiples of each other) but never meet. Skew lines have different directions and also never meet. The key test is comparing direction vectors: if they are proportional, the lines are parallel; if not, and the lines don't intersect, they are skew.

Skew Lines vs. Parallel Lines

Both skew lines and parallel lines never intersect. The difference is that parallel lines share the same direction and lie in a single plane, while skew lines point in different directions and do not share any common plane. Parallel lines can exist in 2D or 3D, but skew lines can only exist in 3D or higher dimensions.

Why It Matters

Skew lines appear frequently in real-world geometry, such as in architecture, engineering, and computer graphics, where structures and paths occupy different planes. Understanding skew lines helps you classify all possible relationships between two lines in 3D space: they must be either intersecting, parallel, or skew. This classification is essential in fields like robotics and 3D modeling, where determining the shortest distance between two non-intersecting paths is a practical problem.

Common Mistakes

Mistake: Assuming two lines that don't intersect must be parallel.
Correction: This is only true in 2D. In 3D, non-intersecting lines can also be skew. Always check whether the direction vectors are proportional before concluding the lines are parallel.
Mistake: Trying to find skew lines in a flat (2D) plane.
Correction: Skew lines cannot exist in two dimensions. If a problem is set in 2D, the lines are either parallel or intersecting — there is no third option.

Related Terms