Lecture on Vectors in Multiple Dimensions

Introduction to Dimensions

• Previously dealt with one-dimensional problems (forward/backward or up/down)
• Extending concepts to two dimensions and more
• Classical mechanics usually considers up to three dimensions
• Two-dimensional vectors: magnitude and direction

Understanding Two-Dimensional Vectors

• Vector Definition: Has both magnitude and direction
• Visual Addition of Vectors:
  • Vector A: specified by arrow length and direction
  • Vector B: same concept
  • Vectors can be shifted without changing identity (magnitude and direction remain constant)
  • Addition Example:
    • Place the tail of Vector B at the head of Vector A
    • The resulting vector from A's tail to B's head is the sum (Vector C)
    • Used in displacement vectors to determine total displacement

Breaking Down Vectors into Components

• Vectors can be expressed as a sum of vertical and horizontal components
• Example: Vector X
  • Expressed as the sum of green vector (vertical) and red vector (horizontal)
  • Allows decomposition into X_vertical and X_horizontal
• Importance:
  • Simplifies two-dimensional problems into two one-dimensional problems

Mathematical Breakdown of Vector Components

• Example Vector A with length 5
  • Angle with positive x-axis: 36.8699 degrees
  • Objective: Find horizontal and vertical components
• Right Triangle Formation:
  • Hypotenuse: magnitude of Vector A

Trigonometric Approach

• Finding Vertical Component:
  • Use sine function: sin(36.8699) = opposite/hypotenuse
  • Calculate: 5 * sin(36.8699) = vertical component magnitude
• Finding Horizontal Component:
  • Use cosine function: cos(36.8699) = adjacent/hypotenuse
  • Calculate: 5 * cos(36.8699) = horizontal component magnitude

Example and Results

• Resulting Components:
  • Vertical (A_y): magnitude of 3
  • Horizontal (A_x): magnitude of 4
• 3-4-5 Pythagorean Triangle
• Significance:
  • Breaks complex vector problems into simpler, one-dimensional problems
  • Example in velocity: breaking a 5 m/s vector into components of 3 m/s (vertical) and 4 m/s (horizontal)