Lecture Notes: Adding Vectors

Introduction to Vectors

• Definition: A vector is a quantity that has both magnitude and direction.
• Example: A force vector of 100 newtons directed east.
  • Magnitude: 100 newtons
  • Direction: East

Adding Parallel Vectors

• Example 1: 100 newtons east + 50 newtons east
  • Resultant: 150 newtons east
• Rule: Directly add magnitudes if vectors are parallel.

Adding Anti-Parallel Vectors

• Example 2: 200 newtons east + 120 newtons west
  • Resultant: 80 newtons east (net force is directed east as 200 > 120)
• Example 3: 60 newtons east + 90 newtons west
  • Resultant: 30 newtons west

Adding Perpendicular Vectors

• Example 4: 80 newtons north + 120 newtons south
  • Resultant: 40 newtons south
• Example 5: 30 newtons east + 40 newtons north
  • Resultant Magnitude: Use the Pythagorean theorem.
    • Formula: [ \text{Resultant} = \sqrt{F_1^2 + F_2^2} ]
    • Magnitude: 50 newtons (familiar 3-4-5 triangle)
  • Direction: Use inverse tangent.
    • Formula: [ \theta = \tan^{-1} \left(\frac{F_y}{F_x}\right) ]
    • Angle: 53.1 degrees relative to x-axis

Example of Vectors in Different Quadrants

• Example 6: 50 newtons west + 120 newtons south
  • Magnitude: 130 newtons (use Pythagorean theorem)
  • Direction:
    • Reference angle: [ \theta = \tan^{-1} \left(\frac{120}{50}\right) = 67.4 \text{ degrees} ]
    • Relative to x-axis: 247.4 degrees (since it is in quadrant III)

Example with Non-Parallel/Perpendicular Vectors

• Example 7: 45 newtons east + 60 newtons south
  • Magnitude: 75 newtons
  • Direction:
    • Reference angle: [ \theta = \tan^{-1} \left(\frac{60}{45}\right) = 53.1 \text{ degrees} ]
    • Resultant force vector in quadrant IV
    • Angle relative to x-axis: 306.9 degrees

General Strategy for Adding Vectors

• When vectors are not parallel/perpendicular:
  • Decompose into Components: Use cosine for x-component, sine for y-component.
  • Sum Components: [ F_x = \sum F_{xi}, F_y = \sum F_{yi} ]
  • Resultant Vector:
    • Magnitude: [ \sqrt{F_x^2 + F_y^2} ]
    • Direction: [ \theta = \tan^{-1} \left(\frac{F_y}{F_x}\right) ]

Example of Component Method

• Vectors: 100 newtons east + 150 newtons at 30 degrees above x-axis
  • Components:
    • [ F_{1x} = 100 \cos(0), F_{1y} = 100 \sin(0) ]
    • [ F_{2x} = 150 \cos(30), F_{2y} = 150 \sin(30) ]
  • Sum of Components:
    • [ F_x = 229.9, F_y = 75 ]
  • Resultant:
    • Magnitude: 241.8 newtons
    • Direction: 18.1 degrees relative to x-axis

Quadrant Rules for Angles

• Quadrant I: Angle = Reference Angle
• Quadrant II: Angle = 180 - Reference Angle
• Quadrant III: Angle = 180 + Reference Angle
• Quadrant IV: Angle = 360 - Reference Angle