Lecture Notes: Introduction to Vectors

Objectives

• Define the term "vector"
• Show how vectors are represented
• State the types of vectors
• Write the component form of a vector
• Discuss equal and opposite vectors

What is a Vector?

• Definition in Mathematics: A quantity with both size (magnitude) and direction (angle).
• Scalar vs. Vector Quantities:
  • Scalar Quantities: Have only magnitude (e.g., mass, length, area, volume).
  • Vector Quantities: Have both magnitude and direction (e.g., velocity, gravitational pull, bearing, force).

Representation of Vectors

• Notation: Represented by two capital letters with an arrow above, showing direction (e.g., ( \overrightarrow{AB} )).
• Column Matrix Representation: Used to denote the components of a vector (e.g., ( \begin{bmatrix} x \ y \end{bmatrix} )).

Types of Vectors

• Position Vectors: Start at the origin (0,0), named with "O" (e.g., ( \overrightarrow{OA} )).
• Free Vectors: Not connected to the origin, named without "O" (e.g., ( \overrightarrow{BC} )).
• Zero Vectors: No magnitude, any direction.
• Unit Vectors: Magnitude of one unit.

Component Form of a Vector

• Notation: Written as a column matrix ( \begin{bmatrix} x \ y \end{bmatrix} ).
• Example:
  • For position vector ( \overrightarrow{AB} ), directly use endpoint coordinates.
  • For free vector ( \overrightarrow{CD} ), calculate as endpoint minus starting point.

Equal and Opposite Vectors

• Equal Vectors: Have the same component form.
• Opposite Vectors: Components add up to zero; signs are opposite.
• Example:
  • ( \overrightarrow{AB} = \begin{bmatrix} 2 \ 5 \end{bmatrix} ) equal to ( \overrightarrow{OM} = \begin{bmatrix} 2 \ 5 \end{bmatrix} ).
  • Opposite if ( \overrightarrow{AB} = \begin{bmatrix} -2 \ -5 \end{bmatrix} ) and ( \overrightarrow{OM} = \begin{bmatrix} 2 \ 5 \end{bmatrix} ).

Future Topics

• Magnitude and direction of vectors
• Vector algebra
• Proving parallelism
• Collinearity of points using vectors

Note: This lecture is an introduction to the concepts of vectors, more detailed explanations and calculations will follow in future sessions.