Vectors

Basics of Vectors:

Definition:

  • Magnitude and Direction: Vectors are quantities defined by both magnitude (size) and direction.
  • Representation: Typically represented as arrows in space, with length indicating magnitude and direction showing its orientation.

Notation:

  • Representation: Vectors are denoted by symbols with an arrow on top (v, u, etc.).
  • Components: Vectors can be represented by their components in various dimensions (2D, 3D, etc.).

 

Types of Vectors:

  • Position Vectors: Represent the position of a point in space relative to a reference point or origin.
  • Displacement Vectors: Represent the change in position from one point to another.
  • Force, Velocity, Acceleration: Common physical vectors representing force, speed with direction, and change in velocity, respectively.

Vector Operations:

Addition and Subtraction:

  • Graphical Method: Vectors are added or subtracted by placing them tip-to-tail or by using the parallelogram law.
  • Component Method: Add or subtract corresponding components of vectors.

Scalar Multiplication:

  • Scaling: Multiplying a vector by a scalar changes its magnitude without altering its direction.

Vector Properties:

  • Commutative Property: v+u=u+v
  • Associative Property: v+(u+w)=(v+u)+w
  • Distributive Property: a(u+v)=au+av
  • Magnitude: The length of the vector, often represented as v.
  • Unit Vector: A vector with a magnitude of 1 in a specific direction.
  • Zero Vector: A vector with zero magnitude; all its components are zero.
  • Parallel Vectors: Have the same or opposite directions, even if their magnitudes differ.
  • Orthogonal (Perpendicular) Vectors: Their dot product is zero.

 

Magnitude and Direction:

  • Magnitude: The length or size of a vector, denoted by vor v.
  • Unit Vector: A vector with a magnitude of 1 in the same direction as the original vector.

 

Dot Product (Scalar Product):

  • Definition:
  • vu=vucos(θ), where θ is the angle between the vectors.
  • Properties: Commutative, distributive, and follows trigonometric rules.

 

Cross Product (Vector Product):

  • Definition: v×u=vusin(θ)n^, where n^ is the unit normal vector perpendicular to the plane formed by v and u.
  • Result: A vector perpendicular to both original vectors.

Applications:

  • Physics: Forces, moments, torque, and electromagnetism.
  • Engineering: Structural analysis, mechanical systems.
  • Computer Graphics: Manipulating objects in 3D space, lighting effects.