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CIE A-Level Maths Study Notes

2.7.5 Parallel, Intersecting, and Skew Lines

Vector analysis is essential for determining the spatial relationships between lines in both two and three-dimensional geometries. This section examines the conditions and methods for identifying parallel, intersecting, and skew lines, providing a step-by-step approach to understanding their interactions.

Parallel Lines

Parallel lines are two or more lines in a plane that never intersect. They have the same slope but different y-intercepts in 2D geometry.

parallel vectors

Image courtesy of Cuemath


Consider two lines on a plane, L1:y=2x+3L1: y = 2x + 3 and L2:y=2x4L2: y = 2x - 4.


1. Slopes: From y=mx+by = mx + b, slope m=2m = 2 for both L1L1 and L2L2.

2. Comparison: Equal slopes mean L1L1 and L2L2 are parallel.

Conclusion: L1L1 and L2L2 are parallel due to identical slopes.

Intersecting Lines

Intersecting lines cross at a single point. In 2D geometry, this occurs when two lines have different slopes.

Intersection of two line using vector

Image courtesy of Wolfram Demostration Project


Two lines, L3:y=x+1L3: y = x + 1 and L4:y=2x+3L4: y = -2x + 3.


1. Equation: Set x+1=2x+3x + 1 = -2x + 3 to find the intersection.

2. Solve for xx: Rearrange to 3x=23x = 2, so x=23.x = \frac{2}{3}.

3. Solve for yy: Substitute xx into L3L3 to get y=53y = \frac{5}{3}.

Conclusion: L3L3 and L4 L4 intersect at (23,53)\left(\frac{2}{3}, \frac{5}{3}\right).

Skew Lines

Skew lines are lines that do not intersect and are not parallel, usually found in 3D geometry.

skew lines

Image courtesy of Cuemath


Consider two lines in 3D space, L5:(x,y,z)=(1,2,3)+t(1,0,1)L5: (x, y, z) = (1, 2, 3) + t(1, 0, 1) and L6:(x,y,z)=(2,4,5)+s(0,1,1)L6: (x, y, z) = (2, 4, 5) + s(0, 1, 1), where tt and ss are parameters.


  1. Direction Vectors: L5L5 has (1,0,1) (1, 0, 1), L6L6 has (0,1,1)(0, 1, 1).
  2. Not Parallel: Direction vectors aren't scalar multiples.

3. No Intersection: No tt and ss satisfy both equations simultaneously.

Conclusion: L5L5 and L6L6 are skew, meaning they do not intersect or run parallel.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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