Lecture on Vectors

Introduction to Vectors and Scalars

• Scalar Quantity: Only has magnitude (e.g., speed, temperature, mass).
  • Examples:
    • Speed: 40 m/s (only magnitude)
    • Temperature: 80 degrees (no direction)
    • Mass: 50 kg (no direction)
• Vector Quantity: Has both magnitude and direction (e.g., velocity, force).
  • Examples:
    • Velocity: 40 m/s north (magnitude and direction)
    • Force: 300 N east (magnitude and direction)

Understanding Vectors

• Vector Representation:
  • Directed line segment from initial point (A) to terminal point (B)
  • Can be represented as ( \vec{AB} )
• Components of Vectors:
  • Described by magnitude and angle or by components (x, y)
  • Example: Vector with components (2, 3) can be represented graphically

Points vs. Vectors

• Points are represented with parentheses (e.g., point (3, 4))
• Vectors use inequality symbols (e.g., vector ( \langle 4, 5 \rangle ))

Vector Magnitude and Components

• Magnitude of vector ( v ) calculated using ( \sqrt{v_x^2 + v_y^2} )
• Determining components: difference between terminal and initial points

Equivalent Vectors

• Vectors are equivalent if they have the same magnitude and direction
• Use component form and magnitudes to check equivalency

Adding and Subtracting Vectors

• Graphically add by connecting head of one vector to tail of another
• Subtraction involves adding the negative of a vector

Vector Operations

• Scalar multiplication changes magnitude, not direction
• Adding vectors involves summing each component

Position and Unit Vectors

• Position Vector: Initial point at origin, terminal point determines vector
• Unit Vector: Vector with magnitude of 1
  • Finding unit vector: divide vector by its magnitude

Standard Unit Vectors

• Standard Unit Vectors: ( \mathbf{i}, \mathbf{j}, \mathbf{k} )
  • ( \mathbf{i} ) for x-direction, ( \mathbf{j} ) for y-direction, ( \mathbf{k} ) for z-direction
• Express vectors using standard unit vectors

Unit Circle and Vectors

• Unit Circle: Radius of 1, relates to unit vectors
• Components of unit vector: cosine and sine functions

Vector Magnitude and Angle

• Calculating vector magnitude: ( \sqrt{x^2 + y^2} )
• Angle determination using trigonometry (arc tangent function)
• Quadrant determination affects angle calculations

Resultant Force Vector

• Resultant force is the sum of multiple vectors
• Calculate magnitude and direction of resultant using components

Summary

• Understand the concepts of vectors and scalars
• Know how to represent and calculate with vectors
• Familiarity with unit and standard unit vectors
• Practical applications of vector addition, subtraction, and representation