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Understanding Vectors: Definition, Representation, and Parametric Plotting

This document introduces the concept of vectors, defining them as quantities possessing both magnitude and direction. Unlike scalars, vectors convey more information and can be visually represented as directed line segments. Key points include the criteria for vector equality, along with practical examples illustrating vector definition by tip and tail, parametric line definitions, and operations such as vector addition and the dot product. This foundational knowledge is essential in fields involving physics, engineering, and mathematics.

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Understanding Vectors: Definition, Representation, and Parametric Plotting

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  1. VC.01 Part A Vectors and Parametric Plotting

  2. Definition of a Vector • A vector is a quantity that has both a magnitude and a direction • Vectors encode more information than scalars (magnitude • without direction) • Vectors can be represented in the plane as a directed line • segment (magnitude represented by length, direction • represented by arrowhead) • Two vectors are equal if they have the same magnitude and • direction, but they don’t have to start from the same point

  3. Example 1: Defining a Vector By Its Tip and Tail

  4. Example 2: Defining a Line Parametrically

  5. Example 2: Defining a Line Parametrically

  6. Example 2: Defining a Line Parametrically

  7. Example 2: Defining a Line Parametrically

  8. Example 2: V + W and V – W

  9. Example 2: V + W and V – W

  10. Example 2: V + W and V – W

  11. Example 2: V + W and V – W

  12. Example 3: Dot Product

  13. Example 3: Dot Product

  14. Example 4: Projection/Push-Pull

  15. Example 4: Projection/Push-Pull

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