Lesson 4: 3D Vectors

• Learning Goal

  • Know about the 3-dimensional Cartesian system (3-space)
  • Plot points and position vectors in 3-space
  • Perform various operations to vectors in 3-space
• Success Criteria

  • Plot points and position vectors in 3-space
  • Determine the magnitude of a vector with 3 components
  • Perform various operations (addition, subtraction, scalar multiplication, and dot product) between vectors in 3-space
• Three-Dimensional Cartesian System


  • Points in 3-D space are described using ordered triples of real numbers
  • Coordinate system in 3-D space is formed using three mutually perpendicular number lines: 𝑥-axis, 𝑦-axis, 𝑧-axis
• Plotting a point in 3-space

  Plot the point (−3, 4, 7) and describe the location
• Position Vector in Three-Space

  Example 1: Plot points A(2,3,4) and B(2, −3, −1), draw the corresponding position vector, write the position vector in component form, and as a linear combination of the unit vectors
• Unit Vectors


  • 𝚤̂ (unit vector in the direction of the positive x-axis)
  • 𝚥̂ (unit vector in the direction of the positive y-axis)
  • 𝑘෠ (unit vector in the direction of the positive z-axis)
• Finding the magnitude of a vector in three-space

  For vector 𝑢ሬ⃗ = (𝑎, 𝑏, 𝑐), the magnitude of |𝑢ሬ⃗| = square root of (𝑎^2 + 𝑏^2 + 𝑐^2)
• Magnitude of a Cartesian Vector in Three-Space is the addition of three vectors
• Finding the magnitude of a vector in three-space

  Addition of three vectors
• Operations with 3-D Vectors: 3-D vectors can be combined to determine resultants in a similar way to 2-D vectors
• Vectors in ℝ𝟑 have an additional component, z, but the properties of scalar multiplication, vector addition and subtraction developed for ℝ𝟐 are valid in ℝ𝟑
• Scalar Multiplication in Three-Space

  For any vector 𝑢ሬ⃗ = (𝑢ଵ, 𝑢ଶ, 𝑢ଷ) and any scalar 𝑘 ∈ ℝ, 𝒌𝑢ሬ⃑ is a scalar multiple of 𝑢ሬ⃑
• Vector Addition in Three-Space


  For any vector 𝑢ሬ⃗ = (𝑢ଵ, 𝑢ଶ, 𝑢ଷ) and 𝑣⃗ = (𝑣ଵ, 𝑣ଶ, 𝑣ଷ), 𝑢ሬ⃑ + 𝑣⃗
• Vector Subtraction in Three-Space


  For any vector 𝑢ሬ⃗ = (𝑢ଵ, 𝑢ଶ, 𝑢ଷ) and 𝑣⃗ = (𝑣ଵ, 𝑣ଶ, 𝑣ଷ), 𝑢ሬ⃑ − 𝑣⃗
• Vector Between Two Points: The vector 𝑃𝑄ሬሬሬሬሬ⃗ from point 𝑃(𝑥ଵ, 𝑦ଵ, 𝑧ଵ) to the point 𝑄(𝑥ଶ, 𝑦ଶ, 𝑧ଶ) is...
• The magnitude of 𝑃𝑄ሬሬሬሬሬ⃗ is |𝑃𝑄|ሬሬሬሬሬሬሬሬሬ⃗ = √((𝑥ଶ − ��ଵ)² + (𝑦ଶ − 𝑦ଵ)² + (��ଶ − 𝑧ଵ)²)
• Point to point vector


  𝑃𝑄 = (𝑥ଶ − 𝑥ଵ, 𝑦ଶ − 𝑦ଵ, 𝑧ଶ − 𝑧ଵ)
• Magnitude of a vector


  |𝑃𝑄| = √((𝑥ଶ − 𝑥ଵ)² + (𝑦ଶ − 𝑦ଵ)² + (𝑧ଶ − 𝑧ଵ)²)
• Scalar multiple
• Proof
• Given vectors

  • 𝑢ሬ⃑ = (2,3, −5)
  • 𝑣⃑ = 8��̂ − 3𝑘෠
• Given points


  • 𝐴(3,6, −1)
  • 𝐵(1,5,2)
• Recall... 𝑢ሬ⃑ ⋅ 𝑣⃑ = 0 when…
• Angle between vectors


  • 𝑢ሬ⃑ = (−3,4,1)
  • 𝑣⃑ = (2,5,3)
• The Properties of Addition and Scalar Multiplication and the Dot Product are the same in Two-Space as it is in Three-Space
• Homework exercises