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Introduction to Vectors

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Agenda

  • What are Vectors?
  • Application of Vectors
  • Practice

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What are Vectors?

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What are Vectors?

  • Calculus is the mathematics of change
  • Vectors is the mathematics of things that move
  • These sound very similar
    • Are they the same?
    • Not quite!
  • Vector Calculus is a thing
    • But not until 2nd year university 😳

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What are Vectors?

  • Vectors describe moving things
  • But with vectors we tend not to be worried about the rate of change (or slope)
    • At least not yet!
    • And if we do (when calculating velocity) then we simplify it to remove unnecessary details

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Example: Video Games

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Example Reinforcement Learning

  • Reinforcement Learning Video

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How Can We Describe This?

  • For the objects in the videos
  • Don’t worry about the details
  • Focus on how they move
    • Start at a location
    • End at a location
  • In its simplest form this is
    • How far is the movement
    • What is the direction from start to finish

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Vector Definition

  • A vector has a length and a direction
  • Vectors can move anywhere and are still the same vector
  • In the picture there are two different vectors
  • Vectors are written with a name or are given by the endpoints from start to finish

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Magnitude of a Vector

  • The magnitude (length) of a vector is written

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  • This is frequently calculated using Pythagorean Theorem or trig

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Opposite Vectors

  • If two vectors have the same magnitude (length) but opposite direction they are called opposite vectors
  • One is written as the negative of the other

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Equivalent Vectors

  • If two vectors have the same magnitude and direction then they are equal (or equivalent) vectors
  • Parallel vectors with the same length are equal

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Example

  • Look at the Jamboard V1 Examples for equivalent and opposite vectors

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Application of Vectors

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Where are Vectors Used?

  • Anywhere that there is movement, there is (usually) an application of vectors
  • A plane’s motion through the air
  • A runner’s path
  • A subatomic particle’s motion in a collider

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Where are Vectors Used?

  • Vectors can also be used in more abstract settings
  • The force one object exerts on another
  • A list of information
    • See 3Blue1Brown house example
    • Stock trades for a day

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Interpretations of Vectors

From 3Blue1Brown

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Practice

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Example

  • Look at the Jamboard V1 Examples

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Credits

Special thanks to all the people who made and released these awesome resources for free:

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  • Presentation template by SlidesCarnival

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