M408M Learning Module Pages
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Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space

Learning module LM 12.1: 3-dimensional rectangular coordinates:

Learning module LM 12.2: Vectors:

Learning module LM 12.3: Dot products:

       Definitions
      Properties
      Projections and components
      Worked problems

Learning module LM 12.4: Cross products:

Learning module LM 12.5: Equations of Lines and Planes:

Learning module LM 12.6: Surfaces:

Chapter 13: Vector Functions

Chapter 14: Partial Derivatives

Chapter 15: Multiple Integrals

Worked problems

Worked problems

In the following video, we work through two problems:

  1. Find the angles of a triangle with vertices at $P(2,-2,-1)$, $Q(3,1,-3)$ and $R(7,-1,-4)$.
  2. Find the angle between the diagonals of two faces of a cube, if those diagonals meet at a vertex.