Vectors and the Geometry of Space: Equations of Lines and Planes

in DTubelast year (edited)

In this video I go over equations of lines and planes in 3D coordinates. This section is part of my calculus book chapter titled Vectors and the Geometry of Space. In 2D coordinates, a line can be described by a point on the line and its slope or direction. Similarly, a line in 3D coordinates can be described by a point on the line and a vector parallel to it which thus serves as the direction. A plane in 3D coordinates is described by a point on the plane and a vector perpendicular to the plane. The plane equation utilizes the fact that the dot product of two perpendicular vectors is equal to zero. Furthermore, the cross product is also used in determining the perpendicular vector, termed a normal vector. I go through several related concepts, such as scalar and vector equations of lines, segments of lines, and linear equations of planes, as well as many examples to give a thorough examination of 3D lines and planes.

The topics covered as well as their timestamps are listed below.

  • Introduction: 0:00
  • Calculus Book Reference: 1:13
  • Sections in Calculus Book Chapter: 1:27
  • Topics to Cover: 2:28
  1. Lines: 3:42
    • Vector Equation of a Line: 5:00
    • Parametric Equations of a Line: 12:50
    • Example 1: 17:22
      • Solution to 1(a): 17:49
      • Solution to 1(b): 25:55
    • Direction Numbers: 28:08
    • Symmetric Equations of a Line: 31:49
    • Example 2: 37:29
      • Solution to 2(a): 37:54
      • Solution to 2(b): 43:43
      • Note on Example 2: 48:38
    • Equation of a Line Segment: 50:24
    • Example 3: 1:04:16
  2. Planes: 1:15:15
    • Vector Equation of a Plane: 1:15:32
    • Scalar Equation of a Plane: 1:21:13
    • Example 4: 1:23:14
    • Linear Equation of a Plane: 1:29:40
    • Example 5: 1:32:58
    • Example 6: 1:42:27
    • Parallel Planes: 1:48:18
    • Example 7: 1:52:34
      • Solution to 7(a): 1:52:58
      • Solution to 7(b): 1:58:13
    • Note 1: Two Linear Equations Can Represent a Line: 2:08:20
    • Note 2: Line of Intersection from Two Parametric Planes: 2:16:18
    • Example 8: Distance from a Point to a Plane: 2:19:55
    • Example 9: 2:30:45
    • Example 10: 2:38:04
    • Exercise 1: 2:55:19

Related Videos:

Discovery Project: The Geometry of a Tetrahedron:

The Cross Product:

The Dot Product:

Complex Numbers as Rotation Matrices:


3D Coordinate Systems: .


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In this video I go over an extensive examination of the equations of lines and planes in 3D coordinates.

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