Concept of a Vector

Concept of a Vector

  • A vector is a quantity that possesses both magnitude (or size) and direction, unlike a scalar which only has magnitude.
  • Vectors are often represented as an arrow with the length indicative of the magnitude and direction shown by the way the arrow is pointing.
  • In a coordinate system, vectors are often described using column vectors or coordinates. The vector (a, b) represents a move a units to the right and b units upwards.
  • Vectors can be added together to find a resultant vector. This is done by ‘placing the vectors head to tail’ and seeing where you end up relative to your starting position.
  • Vector subtraction involves adding the opposite (or negative) of a vector.
  • You can multiply vectors by scalars (a number). This has the effect of changing the magnitude of a vector, but not its direction, unless the scalar is negative, in which case the vector direction is reversed.
  • The zero vector, or null vector, is a vector that has zero magnitude and an arbitrary direction. It is represented as (0,0).
  • Equal vectors have the same length and direction, but not necessarily the same starting point.

Position Vectors

  • The position vector of a point is the vector drawn from the origin to that point. It is generally denoted as r.
  • To calculate a position vector, simply subtract the origin’s coordinates from the position’s coordinates.

Vector Algebra

  • Vectors follow the commutative law, which means the order in which they are added doesn’t change the result.
  • Vectors also obey the distributive law, which means that a scalar multiplied with the sum of two vectors results in the same value as adding the scalar product of each separate vector.

Vector Geometry

  • The dot product (or scalar product) of two vectors results in a scalar quantity. It gives the magnitude of one vector projected onto another.
  • The cross product (or vector product) of two vectors results in another vector. This vector is perpendicular to the plane containing the two original vectors.
  • The angle between two vectors can be found using the dot product, and this concept is key in many geometry problems.