Vector Geometry: Lines and Planes in 3D Space

Defining Lines and Planes in 3D Space

Plane Definitions

• Plane defined by two intersecting lines

  Formed by the direction vectors of both lines and their intersection point.

• Plane through a point, parallel to another plane

  Take the normal vector of the parallel plane and the given point.

• Plane through a point, perpendicular to a line

  The given point and the direction vector of the line will be the normal vector of the plane.

• Plane through a point (x₀, y₀, z₀), perpendicular to two planes

  The normal vector n₃ is the cross product of the normal vectors of the two planes: n₃ = n₂ × n₁. The plane's equation is: n₃ₓ(x - x₀) + n₃ᵧ(y - y₀) + n₃𝓩(z - z₀) = 0.

Line Definitions

• Line through a point, parallel to two planes

  The direction vector vᵣ of the line will be perpendicular to the normal vectors of both planes: vᵣ = n₁ × n₂.

• Line through a point, parallel to a plane, perpendicular to another line

  The direction vector vᵣ of the line will be the cross product of the plane's normal vector and the other line's direction vector: vᵣ = n_plane × v_line.

• Line through a point, perpendicular to two lines

  The direction vector of the desired line will be the cross product of the direction vectors of the two given lines: v = v₁ × v₂.

• Line perpendicular to two skew lines

  The direction vector of this line is the cross product of the direction vectors of the two skew lines. This line can be defined as the intersection of two planes:

  • One plane contains the first line and the direction vector of the perpendicular line.
  • The other plane contains the second line and the direction vector of the perpendicular line.

• Line through a point, perpendicularly intersecting another line

  To find this line, determine the point on the given line that forms a perpendicular segment with the given external point. The direction vector of the desired line will be the vector connecting these two points. The orthogonality condition (dot product of direction vectors is zero) is used to find the specific point on the given line.

Relative Position of Geometric Objects

Relative Position of Two Lines

• Lines defined by a point and a direction vector

  For two lines defined by a point P and a direction vector v, analyze the rank of the matrix formed by the vector connecting the two points (P₂ - P₁) and the two direction vectors (v₁, v₂).

  • If Rank(M) = 3: Lines are skew (do not intersect and are not parallel).
  • If Rank(M) = 2:
    • If v₁ and v₂ are not proportional (v₁ ≠ k ⋅ v₂): Lines intersect.
    • If v₁ and v₂ are proportional (v₁ = k ⋅ v₂) AND the vector (P₂ - P₁) is not proportional to v₁: Lines are parallel.
  • If Rank(M) = 1: Lines coincide (are the same line).

• Two lines defined by their implicit equations

  Let A be the coefficient matrix and A' be the augmented matrix of the system of four equations (two for each line, as intersection of planes).

  • If Rank(A) = 3 and Rank(A') = 4: Lines are skew.
  • If Rank(A) = 3 and Rank(A') = 3: Lines intersect.
  • If Rank(A) = 2 and Rank(A') = 3: Lines are parallel.
  • If Rank(A) = 2 and Rank(A') = 2: Lines coincide.

Relative Position of a Line and a Plane

• Line (Parametric) and Plane (General Form)

  Line: x = x₀ + λu₁, y = y₀ + λu₂, z = z₀ + λu₃.

  Substitute the line's parametric equations into the plane's equation: a(x₀ + λu₁) + b(y₀ + λu₂) + c(z₀ + λu₃) + d = 0.

  • If (a ⋅ u₁ + b ⋅ u₂ + c ⋅ u₃) ≠ 0: The line and plane intersect at a single point.
  • If (a ⋅ u₁ + b ⋅ u₂ + c ⋅ u₃) = 0 AND (a ⋅ x₀ + b ⋅ y₀ + c ⋅ z₀ + d) = 0: The line is contained within the plane.
  • If (a ⋅ u₁ + b ⋅ u₂ + c ⋅ u₃) = 0 AND (a ⋅ x₀ + b ⋅ y₀ + c ⋅ z₀ + d) ≠ 0: The line is parallel to the plane (and not contained within it).

• Line (Implicit) and Plane (General Form)

  Consider the system of three linear equations (two for the line, one for the plane). Let A be the coefficient matrix and A' be the augmented matrix.

  • If Rank(A) = 3 and Rank(A') = 3: The line and plane intersect at a single point.
  • If Rank(A) = 2 and Rank(A') = 3: The line is parallel to the plane.
  • If Rank(A) = 2 and Rank(A') = 2: The line is contained within the plane.

Relative Position of Three Planes

For three planes (a₁x + b₁y + c₁z + d₁ = 0; a₂x + b₂y + c₂z + d₂ = 0; a₃x + b₃y + c₃z + d₃ = 0):

Let A be the coefficient matrix and A' be the augmented matrix.

• If Rank(A) = 3 and Rank(A') = 3: The three planes intersect at a single point (form a trihedral).
• If Rank(A) = 2 and Rank(A') = 3: System is incompatible (no common intersection point).
  • If two normal vectors are not proportional: The planes form a triangular prism (intersect pairwise in parallel lines).
  • If two normal vectors are proportional: Two planes are parallel, and the third plane intersects them.
• If Rank(A) = 2 and Rank(A') = 2: System is compatible indeterminate. The planes intersect in a common line.
• If Rank(A) = 1 and Rank(A') = 2: System is incompatible. Three planes are parallel (at least two distinct).
• If Rank(A) = 1 and Rank(A') = 1: System is compatible indeterminate. The three planes coincide (are the same plane).