Checklist for Vectors

Many schools have been doing vectors recently. Thought I’ll share a little summary/ checklist I have done for my students.

Basic Concepts

  • Operations on Vectors
    • Addition & Subtraction
    • Scalar multiplication
    • Dot Product (Scalar)
      1. a • a = |a|2
      2. If a ⊥ b, then a • b = 0
      3. a • b = b • a
    • Cross Product (Vector)
      1. a × a = 0
      2. a × b = − b × a
  • Unit Vectors
  • Parallel Vectors ( a = k)
  • Collinear Vectors ( Parallel with a common point )
  • Ratio Theorem ( Found in MF26)
    • Midpoint Theorem
  • Directional Cosines
  • Angles between two Vectors
  • Length of Projection
  • Perpendicular Distance

Lines

  • Equations
    • Vector Form ( : r = a + λb, λ∈ ℜ )
    • Parametric Form
    • Cartesian Form
  • Line & Line
    • Parallel ( Directions are parallel to each other. )
    • Same ( Same Equations )
    • Intersecting ( There is a unique solution for λ and μ. )
    • Skewed ( Not parallel AND not Intersecting. )
  • Angle between two lines ( Angle between their directions )
  • Point & Line
    • Foot of Perpendicular
    • Perpendicular (Shortest) distance
    • Point on Line

Planes

  • Equations
    • Parametric Form ( π r = a + λb + μc, λ, μ ∈ ℜ )
    • Scalar Product Form ( r • n = a • n  = d )
    • Cartesian Form
  • Point & Plane
    • Foot of Perpendicular
    • Perpendicular (Shortest) distance
    • Distance from O to Plane
    • Point on Plane
    • Reflection of Point
  • Line & Plane
    • Relationships
      1. Parallel
        • Line intersects Plane entirely ( Infinite Solutions )
        • Do not intersect ( No Solution )
      2. Not Parallel
        • Intersects at a point ( One Solution )
    • Intersection Point
    • Angle between Line & Plane
    • Reflection of Line
  • Plane & Plane
    • Relationships
      1. Parallel
        • Same ( Infinite Solutions )
        • Do not intersect ( No Solution )
      2. Not Parallel
        • Intersects at a line ( Infinite Solutions )
    • Intersection Line ( Use of GC )
    • Angle between two Planes ( Angle between their normals )

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