Shortest distance between two skew lines

Shortest Distance between Two Skew Lines Overview

  • Skew lines are a pair of lines in three-dimensional space that do not intersect and are not parallel.
  • The shortest distance between two skew lines is the shortest line segment that can be drawn from one line to the other.
  • This concept is central to vector geometry.

Key Terms for Skew Lines

  • Directions ratios of a line are the coefficients of i, j, k in the directional vector of the line.
  • The directional vector of the line is a vector that gives the direction of the line.
  • Cross product of two vectors results in a vector that is perpendicular (orthogonal) to both of the original vectors.

Finding the Shortest Distance

  • The shortest distance between two skew lines can be found using the cross product of their directional vectors.
  • The formula for the shortest distance between two skew lines is: D = (b1 - a1) . (a2 × b2) / a2 × b2
    • here . represents scalar product
    • and × represents vector product
  • The result is the absolute value of the dot product of the vector connecting a point on one of the lines to a point on the other line, and the cross product of the directional vectors of the lines, all over the magnitude of the cross product of the directional vectors.

Applications of Shortest Distance between Skew Lines

  • The concept of shortest distance between skew lines is frequently used in physics, engineering and computer graphics.
  • It aids in solving problems involving three-dimensional geometry and understanding spatial relationships.

Understanding through Examples

  • Example: Consider the skew lines with parametric equations x = t, y = 2t, z = 3t and x = 2 + s, y = 4 + 2s, z = 6 + 4s. The shortest distance between these two lines can be determined using the formula mentioned above.

Potential Challenges

  • Understanding how to execute cross and dot products.
  • Grasping the concept of skew lines, particularly visualising them in three dimensions.
  • Applying the formula correctly, particularly ensuring directional vectors are accurately determined before proceeding.

In-depth understanding of shortest distances between skew lines and effective application of related formulae are integral to mastering vector geometry, a fundamental area within the Further Pure 1 module.