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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

Example:

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

Solution:

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

Example:

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

Solution:

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

Example:

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.

Solution:

  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
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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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