Three Dimensional Geometry

Notes on Three Dimensional Geometry

Introduction

Three-dimensional geometry extends the concepts learned in two-dimensional analytical geometry to three dimensions. The main goal is to represent lines and planes using vectors, which simplifies the calculations and provides a cohesive framework for understanding spatial relationships. Vector algebra is particularly useful for calculations involving direction cosines, ratios, and distances in 3D space.

Direction Cosines and Direction Ratios

1. Definition:

   • The direction cosines (l, m, n) of a line are the cosines of the angles made with the x, y, and z axes, respectively.
   • These angles are denoted as α, β, γ and can be calculated as follows:
     • l = cos(α), m = cos(β), n = cos(γ).
   • If the direction of the line reverses, the direction cosines change sign.

2. Uniqueness:

   • A unique set of direction cosines can be determined for a directed line (one that has a specific orientation) through a given point.
   • Direction ratios (a, b, c) are any three numbers proportional to the direction cosines, i.e., a = λl, b = λm, c = λn where λ ≠ 0.
   • There are infinitely many sets of direction ratios for a line.

3. Finding Direction Cosines from Two Points:

   • For two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂), the direction cosines are calculated as:

     l = (x₂ - x₁) / PQ,
     m = (y₂ - y₁) / PQ,
     n = (z₂ - z₁) / PQ,

     where PQ = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].

4. Examples:

   • Compute the direction cosines for lines given specific angle measures or points.
   • Direction ratios of a line can be derived directly from coordinates of two points.

Equations of Lines in Space

1. Vector Equations:

   • A line can be defined by the vector equation r = a + λb, where:
     • r is the position vector of any point on the line.
     • a is the position vector of a known point on the line.
     • b represents the direction vector of the line.

2. Parametric Equations:

   • From the vector form, the parametric equations can be extracted as:
     • x = x₁ + λa,
     • y = y₁ + λb,
     • z = z₁ + λc.

3. Cartesian Equations:

   • The Cartesian equation of the line can be obtained by eliminating λ from parametric forms, resulting in:

     (x - x₁)/a = (y - y₁)/b = (z - z₁)/c.

Angles and Distances

1. Angle Between Two Lines:

   • For two lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂), the angle θ between the two lines can be derived as:
     • cos(θ) = (a₁a₂ + b₁b₂ + c₁c₂) / √(a₁² + b₁² + c₁²)√(a₂² + b₂² + c₂²).

2. Shortest Distance Between Skew Lines:

   • Skew lines are defined as lines that are neither parallel nor intersecting. To find the shortest distance between skew lines:

     • Use the formula involving the cross product of direction vectors and the vector between any points on each line:
     • d = |(b₁ x b₂) • (a₂ - a₁)| / |b₁ x b₂|.

3. Example Problems:

   • Problems demonstrate how to calculate angles, distances, and equations of lines using examples in vector form and parametric applications.

1. Direction Cosines: Cosines of angles with coordinate axes.
2. Direction Ratios: Proportional values to direction cosines.
3. Vector Form: A line can be expressed as r = a + λb.
4. Angle Between Lines: Calculated using direction ratios or cosines.
5. Skew Lines: Lines that do not intersect or run parallel.
6. Shortest Distance: Calculated using cross products perpendicular to skew lines.
7. Cartesian Equations: Derived from vector descriptions of lines.
8. Parametric Equations: Represent relationships of coordinates along a line.
9. Line Relationships: Determine collinearity and parallelism through direction ratios.