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Understanding Vectors and Their Applications

Jun 3, 2025

Vectors Lecture Notes

Introduction to Vectors

  • Vectors: Represent journeys from a start point to an endpoint.
  • Column Vectors: Indicate directions on a grid (e.g. six to the right, three down).
  • Algebraic Representation: Vectors can be represented by bold letters or underlined when handwritten.

Writing Vectors

  • Notations: Use arrows over letters to indicate direction (e.g., ( \overrightarrow{OX} ) = A).
  • Reversing Vectors: Changes the sign (e.g., ( \overrightarrow{XO} ) = -A).

Vector Journeys

  • Combination of Vectors: Use existing vectors to describe journeys (e.g., ( \overrightarrow{XY} ) = -A + B).

Exam-Style Problems

  • Example Problem: Given a diagram, express vectors in terms of given variables (A and B).
  • Finding Vector Paths: Use known vectors to find indirect paths (e.g., ( \overrightarrow{QP} ) = -5A + B).
  • Simplifying Vectors: Collect like terms (e.g., ( \overrightarrow{PR} ) = 4A + 2B).

Parallelograms and Midpoints

  • Parallelogram Properties: Opposite sides have equal vector values.
  • Midpoints: Divide the vector by 2 (e.g., ( \overrightarrow{AM} ) = ( \frac{1}{2} ) ( \overrightarrow{AC} )).
  • Example: ( \overrightarrow{OM} ) = 2A + 3B, where M is midpoint.

Ratios and Proportions

  • Vectors in Ratios: Use given ratios to find parts of vectors (e.g., AX:XB = 3:1).
    • Example: ( \overrightarrow{AX} ) = ( \frac{3}{4} ) ( \overrightarrow{AB} ).

Parallel Vectors

  • Definition: One vector is a multiple of the other.
  • Example: If ( \overrightarrow{CD} = 2 \overrightarrow{AB} ), then ( \overrightarrow{CD} ) is parallel to ( \overrightarrow{AB} ).

Collinearity and Straight Lines

  • Vectors on a Line: Two vectors sharing a common point and being parallel suggest collinearity.
    • Example: If ( \overrightarrow{PQ} ) and ( \overrightarrow{QR} ) share a common point and are parallel, P, Q, R are collinear.

Solving for Variables

  • Equation Setup: Equating coefficients in vectors to solve for unknowns (e.g., (-9 = nK)).
  • Applications: Problems involving vectors being parallel to find unknown values (e.g., finding ( K ) when ( \overrightarrow{AB} ) is parallel to ( \overrightarrow{CD} )).

Conclusion

  • Recap: Understanding vectors through algebraic representation, parallelism, and problem-solving techniques.
  • Further Study: Engage with practice problems for exam preparation.

The lecture concludes with encouragement to explore additional resources and practice exercises provided in the video description.

Full transcript