Vector Addition Methods

Jun 12, 2025

Overview

This lecture explains the laws of addition of vectors, including geometric (triangle), parallelogram, and polygon methods, plus vector subtraction and related example problems.

Geometrical Method of Vector Addition

  • The geometric (triangle) method involves placing vectors head-to-tail; the resultant connects the start of the first to the end of the last.
  • Vector addition is independent of the order (commutative law).
  • Example: The position vector of the midpoint of AB is (A + B)/2.

Parallelogram Law of Vector Addition

  • If two vectors are represented as two adjacent sides of a parallelogram from the same point, the diagonal gives their resultant.
  • Resultant magnitude formula: ( R^2 = P^2 + Q^2 + 2PQ \cos \theta ).
  • Resultant direction: ( \tan \alpha = \frac{Q \sin \theta}{P + Q \cos \theta} ).
  • Special cases:
    • θ = 0°: Vectors in the same direction, resultant is sum.
    • θ = 180°: Vectors opposite, resultant is difference.
    • θ = 90°: Vectors perpendicular, resultant is ( \sqrt{P^2 + Q^2} ).

Polygon Law of Vector Addition

  • When more than two vectors, place them head-to-tail in sequence; resultant connects the start of the first to the end of the last.
  • This method extends triangle law to any number of vectors.

Subtraction of Vectors

  • To subtract B from A, add A to the reversed vector of B (written as -B).
  • Subtraction follows the same head-to-tail arrangement.

Illustrative Examples

  • Example: Sum of two forces at an angle yields resultant by parallelogram law.
  • Example: If doubling a vector results in doubled resultant, the angle between vectors can be calculated.
  • Example: If sum and resultant magnitudes are given, with one at right angle, individual forces can be found.
  • Example: Given unit vectors summing to a unit vector, angle between them is 120°.
  • Example: Problem involving ratios of vectors and resultants using given vector relationships.

Key Terms & Definitions

  • Vector — A quantity with both magnitude and direction.
  • Resultant Vector — The single vector representing the sum of two or more vectors.
  • Parallelogram Law — A rule stating that the sum of two vectors can be represented by the diagonal of a parallelogram.
  • Polygon Law — Extension of the triangle law to more than two vectors.
  • Commutative Law — The result of vector addition does not depend on the order of adding vectors.

Action Items / Next Steps

  • Practice Exercise: Answer conceptual questions about vector resultants and properties.
  • Review solved illustrations for various vector addition scenarios.
  • Practice problems using the parallelogram and polygon laws of vector addition.