Overview
This lecture covers how to add vectors, both in the same line and at various angles, using component methods and trigonometric formulas.
Vector Basics
- A vector has both magnitude (size) and direction.
- Example: 100 N force east means 100 N (magnitude), east (direction).
Adding Parallel and Opposite Vectors
- Add magnitudes directly if vectors are parallel and in the same direction.
- Assign positive/negative signs for direction; subtract magnitudes if vectors are opposite.
- Result’s direction is that of the larger vector.
Perpendicular Vector Addition
- For perpendicular vectors, use the Pythagorean theorem: resultant = √(f₁² + f₂²).
- To find direction, use θ = arctan (y-component/x-component).
Example Problems
- 30 N east + 40 N north → resultant = 50 N at 53.1° above x-axis.
- 50 N west + 120 N south → resultant = 130 N at 247.4° relative to x-axis.
- 45 N east + 60 N south → resultant = 75 N at 306.9° relative to x-axis.
Reference Angles and Quadrants
- Calculate the reference angle using arctan(|y/x|); always between 0° and 90°.
- Adjust for quadrant:
- Quadrant I: angle = reference angle
- Quadrant II: angle = 180° – reference angle
- Quadrant III: angle = 180° + reference angle
- Quadrant IV: angle = 360° – reference angle
Adding Vectors at Any Angle (Component Method)
- Break each vector into x (cos) and y (sin) components:
- x = magnitude × cos(angle)
- y = magnitude × sin(angle)
- Add all x components and all y components to get resultant components.
- Magnitude: result = √(x² + y²)
- Direction: θ = arctan (y/x)
Key Terms & Definitions
- Vector — a quantity with both magnitude and direction.
- Resultant Vector — the sum of two or more vectors.
- Component Method — breaking vectors into x and y parts before adding.
- Reference Angle — acute angle used to find direction relative to axes.
Action Items / Next Steps
- Practice breaking vectors into components and adding.
- Review SOHCAHTOA and arctan functions for finding angles.
- Complete assigned problems on vector addition from textbook.