General Vectors

General Vectors

What is a Vector?

  • A vector is a quantity that has both magnitude (a length) and direction. It is usually represented by a line segment directed from one point to another.
  • Unlike scalars, which only have magnitude, vectors account for both how much and where.
  • Vectors can exist in any number of dimensions; however, one, two, or three dimensions are typical in A Level mathematics.

Vector Notation

  • Vectors are commonly denoted by lowercase boldface letters such as a, b, or v.
  • The notation used to denote the vector from point A to point B is often AB.
  • Alternatively, vectors can be represented with i, j, k components. For example, a vector in three dimensions can be depicted as ai + bj + ck.

Vector Arithmetics

  • Addition of Vectors: To add two vectors, you ‘place them head to tail’ and draw the resultant vector from the tail of the first vector to the head of the second.
  • Subtraction of Vectors: To subtract a vector, you simply add its negative or ‘opposite’. This involves reversing the direction of the vector you are subtracting and then adding.
  • Multiplication by Scalars: When a vector is multiplied by a scalar, the magnitude of the vector changes but the direction remains the same (unless the scalar is negative, which reverses the direction).

Unit Vectors

  • A unit vector is a vector of length 1. It typically represents the direction of a vector.
  • i, j and k are the standard unit vectors in the x, y and z directions respectively.
  • Any vector can be converted to a unit vector by dividing the vector by its own magnitude.

Vector Applications

  • Vector concepts are used in a variety of fields, including physics, engineering, computer graphics and navigation.
  • In physics, for example, vectors are useful for representing velocity, force and displacement.
  • In areas like computer graphics, vectors are necessary to render three-dimensional images.