Basic Concepts of Vectors

  • Vectors can be expressed in terms of their components. A two-dimensional vector, for example, can be expressed as (x, y), while a three-dimensional vector can be described as (x, y, z).
  • Magnitude of a vector: The size or length of a vector is referred to as the magnitude, represented as |v|. It can be calculated using the Pythagorean theorem.

Vector Operations

  • Vector addition: Vectors are added by adding corresponding components. If a = (x1, y1) and b = (x2, y2), then a + b = (x1 + x2, y1 + y2).
  • Scalar multiplication: A vector can be multiplied by a scalar (a number). If a = (x, y) and k is a scalar, then ka = (kx, ky).

Vector Applications

  • Position vectors: This identifies a point in space relative to an origin. If O is the origin and P is the point (x, y), the position vector OP = (x, y).
  • Displacement vectors: This describes movement from one point in space to another.

Properties of Vectors

  • Vectors are equal if they have the same magnitude and direction, regardless of their initial point.
  • The zero vector, denoted by 0 or 0**, has a magnitude of zero and is undefined .

Remember, vectors represent quantities with both magnitude and direction. They are used to describe various mathematical and physical concepts, including force, velocity, and displacement.