Lecture on Vectors

Scalar vs Vector Quantities

• Scalar Quantity:
  • Has magnitude only.
  • Example: Mass (e.g., 30 kg).
• Vector Quantity:
  • Has magnitude and direction.
  • Example: Force (e.g., 200 N east).

Examples of Quantities

• Scalar Quantities:
  • Speed
  • Temperature
• Vector Quantities:
  • Velocity (speed with direction)
  • Acceleration

Understanding Vector Components

• Components of a Vector:
  • Represented in terms of i and j.
  • Example: A vector from point A(1,2) to B(4,6).
  • Components: 3i + 4j

Finding Magnitude of a Vector

• Use Pythagorean theorem:
  • Formula: ( v = \sqrt{v_x^2 + v_y^2} )
  • Example: Magnitude of vector (3i + 4j) is 5.

Vector Calculations

• Example 1:

  • Points A(-4,-5) to B(1,7)
  • Vector V: 5i + 12j
  • Magnitude: 13

• Example 2:

  • Initial Point A(-1,-3), Vector V: 3i - 2j
  • Terminal Point B: (2,-5)

Using Vector Components to Find Points

• Finding Terminal Point:

  • Use the formula ( V = (x_2-x_1)i + (y_2-y_1)j )
  • Example: Initial Point A(2,3), Vector V: 5i - 4j
  • Terminal Point B: (7,-1)

• Finding Initial Point:

  • Given terminal point and vector, solve for initial point.
  • Example: Vector V = -3i + 4j, Terminal B(5,3)
  • Initial Point A: (8,-1)

3D Vectors

• Calculate vector V using x, y, z components.
• Example:
  • Points: A(3,4,-2) to B(2,8,3)
  • Vector V: -1i + 4j + 5k
  • Magnitude: ( \sqrt{1^2+4^2+5^2} = \sqrt{42} )

Vector Magnitude and Direction

• Magnitude:
  • Formula: ( \sqrt{x^2 + y^2} )
  • Example: Vector (4i + 6j), Magnitude: ( 2\sqrt{13} )
• Direction:
  • Angle ( \theta ) relative to the x-axis
  • Use ( \theta = \text{tan}^{-1}(v_y/v_x) )
  • Example: Angle = 56.3 degrees