Simplified Minds Lecture on Direction Cosines and Line Equations in 3D Space

Introduction

• Lecturer emphasized simplicity of the topic.
• Key topics: direction cosines, direction ratios, equations of lines in space, angles between lines, and distance between lines.

Direction Ratios and Direction Cosines

Direction Ratios

• Given a vector: a = 2i + 2j + 2k (or generally 2i + 3j + 4k)
• Components along x, y, z are Direction Ratios (DRs): 2, 3, 4

Direction Cosines

• Angles made by the line with x, y, z axes: α, β, γ
• Direction Cosines (DCs): cos(α), cos(β), cos(γ)
• Represented as L, M, N
• Calculation:
  • L = x/sqrt(x^2 + y^2 + z^2)
  • M = y/sqrt(x^2 + y^2 + z^2)
  • N = z/sqrt(x^2 + y^2 + z^2)

Equation of a Line in Space

Unique Line Through Points

• Need a point and a direction or two points for a unique line.
• Line passing through given point (x1, y1, z1) and parallel to a given vector (a, b, c) has the form:
  • r = a + λd
  • (x - x1)/a = (y - y1)/b = (z - z1)/c = λ

Examples

1. Find the equation of a line passing through (5, 2, -4) and parallel to (3, 2, -8):

• x = 5 + 3λ
  • y = 2 + 2λ
  • z = -4 - 8λ
  • Parametric form: x - 5/3 = y - 2/2 = z + 4/8

Angle Between Two Lines

Concept

• Angle between direction vectors is the same as the angle between lines.
• Use the dot product to find the cos(θ):
  • cos(θ) = (B1 · B2) / (|B1| * |B2|)*

Example

• Two lines with direction vectors (1, 2, 2) and (3, 2, 6):
  • Dot product: 1*3 + 2*2 + 2*6 = 3 + 4 + 12 = 19
  • Magnitudes: |B1| = √(1^2 + 2^2 + 2^2) = √9 = 3, |B2| = √(3^2 + 2^2 + 6^2) = √49 = 7
  • cos(θ) = 19/21
  • θ = cos^(-1)(19/21)*

Perpendicular Lines

• B1 dot B2 should be zero if they are perpendicular.
• Example: Show that vectors (7, -5, 1) and (1, 2, 3) are perpendicular.

Distance Between Two Lines

Concept of Skew Lines

• Lines that are non-parallel and do not intersect.
• Shortest distance: perpendicular distance between them.
• Use cross product and dot product:
  • D = |(A2 - A1) · (B1 x B2)| / |B1 x B2|

Example

• Find the shortest distance between lines:
  • Points: A1 = (1, -1, 1), A2 = (2, 1, -1)
  • Direction vectors: B1 = (2, -1, 1), B2 = (3, -5, 2)
  • Cross product: B1 x B2 = (3, -1, -7)
  • Dot product: (A2 - A1) · (B1 x B2) = (1, 2, -2) · (3, -1, -7)
  • Resulting distance: 10/√59

Summary

• Direction cosines and ratios help in understanding line directions in space.
• Unique line equations can be determined given points and directions.
• Angles and distances between lines are calculable using dot and cross products.
• Understanding these geometric concepts is crucial for more advanced studies in 3D geometry.