Lecture Notes: Vectors and Scalars

Introduction

• Discussion on vectors and scalars from the Kinematics and Dynamics chapter.
• Vectors: Numbers with both magnitude and direction.
  • Examples: displacement, velocity, acceleration, force.
• Scalars: Numbers with magnitude only.
  • Examples: distance, speed, energy, pressure, mass.

Understanding Vectors and Scalars

• Vectors: Represented by arrows.
  • Direction of arrow indicates direction.
  • Length of arrow is proportional to magnitude.
  • Notations: Arrow over quantity or boldface.

Vector Calculations

Vector Addition and Subtraction

• Resultant Vector: Sum or difference of vectors.
• Tip-to-Tail Method:
  • Place tail of second vector at the tip of the first.
  • Ensure proportional length to magnitude.
• Component Method:
  • Break vectors into horizontal (X) and vertical (Y) components.
  • Sum of X components and Y components gives resultant.

Examples:

• Vectors in the same direction: Simply add magnitudes.
• Opposite direction vectors: Subtract magnitudes.

Working with Components

• Breaking Vectors into Components:
  • Use right triangle relations.
  • X component: V * cos(θ)
  • Y component: V * sin(θ)
• Use Pythagorean theorem for finding resultant magnitude.

Example Calculation:

• Given V = 10 m/s, θ = 30°:
  • X = 5√3 m/s
  • Y = 5 m/s

Vector Subtraction

• Subtract by adding a vector with equal magnitude but opposite direction.

Vector Multiplication

Multiplying by Scalars

• Changes magnitude, direction remains parallel or anti-parallel based on scalar.
• Positive scalar: Same direction.
• Negative scalar: Opposite direction.

Multiplying Vectors

• Dot Product:

  • Results in a scalar.
  • Formula: A · B = |A||B| cos(θ)
  • Example application: Work = force · displacement

• Cross Product:

  • Results in a vector.
  • Formula: |A x B| = |A||B| sin(θ)
  • Use right-hand rule for direction.
  • Example application: Torque.

Right-Hand Rule

• Fingers point in direction of first vector.
• Curl fingers towards second vector.
• Thumb points in direction of resultant vector.

Example of Cross Product

• Vectors A and B at 90° angle.
• Magnitude: 12 (units) in the negative Z direction.

Conclusion

• Next topic: Displacement and Velocity.
• Encouragement for questions and further study.

Summary:

• Understanding the difference between vectors and scalars is crucial.
• Vector math involves addition, subtraction, and multiplication, each with unique rules and applications.
• Right-hand rule and component breakdown essential techniques for vector analysis.