Introduction to Vector Algebra

Key Concepts

• Scalar Quantity

  • Defined by a single value with a unit
  • Only specifies the magnitude
  • Examples: Length, mass, speed

• Vector Quantity

  • Defined by both magnitude and direction
  • Examples: Velocity, acceleration, force

Difference Between Scalar and Vector

• Scalar: Only has magnitude
• Vector: Has magnitude and direction

Basic Concepts of Vectors

• Directed Line

  • A line in 2D/3D space can have two directions
  • Directed line segment is a vector

• Vector Representation

  • Vector AB (or simply vector A) has initial point A and terminal point B
  • Magnitude is the distance between points A and B

Finding the Magnitude of a Vector

• Given points A(2,1) and B(5,4):

  • Vector V = AB = (x2-x1)i + (y2-y1)j
  • Magnitude: ( \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} )

• For vector C with components (2,3,6):

  • ( \sqrt{2^2 + 3^2 + 6^2} = \sqrt{49} = 7 )

Types of Vectors

• Zero or Null Vector

  • Initial and terminal points coincide
  • Magnitude = 0
  • No direction

• Unit Vector

  • Magnitude of 1
  • Denotes direction of a vector
  • Given vector A, unit vector is ( \hat{A} = \frac{\text{vector A}}{\text{magnitude of A}} )

• Parallel Vectors

  • One vector is a scalar multiple of another
  • If scalar is positive, vectors are in the same direction
  • If scalar is negative, vectors are in opposite directions

• Equal Vectors

  • Same magnitude and direction
  • Corresponding components are equal

• Negative Vectors

  • Negative of vector AB is vector BA

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These notes cover the fundamental concepts of vectors in vector algebra, including definitions, properties, and calculations for magnitude.