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 = kb )
• 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 ( l : 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 )