Three-dimensional Vector Geometry (Lines)

Learning Objectives

  • Equation of a Line: Understanding the vector equation of a line using position vectors and direction vectors.

  • Concepts Involving a Point and a Line: Finding projections, foot of perpendiculars, and reflection points in relation to a line.

  • Pair of Lines: Determining relationships (parallel, intersecting, skew) and properties of lines in three-dimensional space.

1. Equation of a Line

  • Vector Equation: A line can be defined by a position vector of a fixed point and a direction vector. The general form is given by:

    r = a + d * λ

    where a is a point on line l with position vector a and d is the direction vector, and λ is a scalar parameter.

  • Parametric Form: The line can also be expressed in parametric form:

    x = a₁ + d₁ * λ,

y = a₂ + d₂ * λ,

z = a₃ + d₃ * λ

  • Cartesian Form: If the components of the direction vector d are not zero, the values of x, y, and z can be solved in terms of λ.

2. Concepts Involving a Point and a Line

  • Foot of the Perpendicular: The perpendicular foot point N from point P to line l is determined as follows:

    1. Set N on line l: r = a + d * λ

    2. Compute the shortest distance: Distance = |P - N|.

  • Reflection of a Point: To reflect a point P across line l, use the midpoint theorem to calculate position vectors accordingly.

3. Concepts Involving a Pair of Lines

  • Line Relationships: Two lines can be:

    • Parallel: If their direction vectors are scalar multiples of each other.

    • Intersecting: If there exist values λ and μ such that: A + λd₁ = B + μd₂.

    • Skew Lines: Lines that are neither parallel nor intersecting.

  • Angle Between Lines: The acute angle θ between lines with direction vectors d₁ and d₂ is defined by the formula:

    cos(theta) = (d₁ · d₂) / (|d₁| * |d₂|)

Example Problems

  • Finding Vector Equations: Given two points A(1, 1, -4) and direction vector d = (1, 2, 3), the line equation becomes:

    r = (1, 1, -4) + λ(1, 2, 3).

  • Finding Intersection Point: For two lines defined as:

    Line 1: (x, y, z) = (1, 2, 3) + λ(2, 3, 1)
    Line 2: (x, y, z) = (-1, 1, 0) + μ(1, -1, 1)

    Find values of λ and μ for intersection.

4. Additional Notes

  • Distance Between Lines: For parallel lines, calculate the distance using the foot of perpendicular methods. Utilize cross-product or projection methods as necessary.

  • Useful Theorems: The Pythagorean theorem can be applied to find distances in space.

  • Reflection: Reflecting lines involves geometric and algebraic principles based on the established line equations.

Conclusion

These notes summarize the essential concepts of vector geometry and line equations in three-dimensional space. To deepen understanding, practice applying these equations.