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.