Vectors Lecture Notes

Introduction to Vectors

• Vector: A quantity with both magnitude and direction.
• Scalar: A quantity with only magnitude.
• Examples:
  • Speed: Scalar (e.g., 40 m/s)
  • Velocity: Vector (e.g., 40 m/s north)
  • Force: Vector (e.g., 300 N east)
  • Temperature: Scalar (e.g., 80°F)
  • Mass: Scalar (e.g., 50 kg)

Understanding Vectors

• Can be visualized as directed line segments.
• Initial Point: Starting point of the vector.
• Terminal Point: Ending point of the vector.
• Represented as AB with an arrow.

Describing Vectors

• Magnitude and Angle
  • Length of vector represents magnitude.
  • Angle indicates direction.
• Components
  • Expressed as (x, y) components.
  • Example: Vector A = (2, 3)
    • X-component = 2
    • Y-component = 3

Distinguishing Points from Vectors

• Points: Represented in parentheses (e.g., (3, 4)).
• Vectors: Represented with angle brackets (e.g., <4, 5>).

Vector Calculations

• Component Form: Subtract initial point coordinates from terminal point coordinates.
• Magnitude: Calculated using Pythagorean Theorem.
• Equivalent Vectors:
  • Same magnitude.
  • Same direction.

Adding and Subtracting Vectors

• Addition: Connect vectors head-to-tail.
• Subtraction: Reverse the direction of the vector to be subtracted.
• Scalar Multiplication: Changes length but not direction.

Position Vectors and Unit Vectors

• Position Vectors: Initial point at the origin.
• Unit Vectors: Magnitude is 1.
  • Standard Unit Vectors: i, j, k (x, y, z directions respectively)

Applications and Problem Solving

• Vector Operations: Adding, subtracting vectors using components or unit vectors.
• Unit Circle Relations: Vectors related to angles and trigonometric functions.
• Resultant Vectors: Sum of multiple vectors, find magnitude and direction.

Example Problems

• Find Component Form: Given initial and terminal points.
• Determine Magnitude: Use square root of sum of squares of components.
• Identify Equivalent Vectors: Compare magnitudes and slopes.

Conclusion

• Vectors are essential in representing quantities with direction.
• Understanding vectors involves components, magnitude, angles, and operations like addition/subtraction.