Magnitude of a Vector

Magnitude of a Vector

  • The magnitude of a vector is a scalar quantity, representing the ‘length’ or ‘size’ of the vector.
  • It is also referred to as the norm or modulus of the vector.
  • The magnitude can be found by measuring the length of the vector arrow in a diagram, but it is most often calculated mathematically.
  • The magnitude of a vector is always a positive value or zero. There is no such thing as a negative magnitude.
  • It is denoted by using vertical bars around the vector, such as v . The vertical bars do not indicate a vector direction but a scalar quantity which only has magnitude.

Calculation of the Magnitude of a Vector

  • The magnitude of a vector can be calculated using Pythagoras’ theorem in two dimensions or its extension in three dimensions.
  • For a two-dimensional vector v = (a,b), the magnitude v is calculated as √(a^2 + b^2).
  • For a three-dimensional vector v = (a,b,c), the magnitude v is calculated as √(a^2 + b^2 + c^2).
  • The magnitude of a vector remains the same, regardless of its direction or orientation in space. Simply, it’s a measure of how long the vector is, not where it’s pointing.

Unit Vectors

  • A unit vector is a vector whose magnitude is exactly 1. It is used to represent direction only, as its magnitude gives no other information.
  • Any vector can be converted into a unit vector by dividing the vector by its magnitude. The resulting vector will point in the same direction as the original, but its length is standardized to 1.
  • The unit vectors in Cartesian coordinates are often denoted as i, j, and k in the x, y, and z directions respectively.

Vector Scalars

  • A scalar is a quantity that has only magnitude and no direction. When a vector is multiplied by a scalar, the magnitude of that vector is scaled by the scalar value.
  • When the scalar is positive, the direction of the vector remains the same. When it is negative, the vector is reversed in direction.
  • If the scalar is less than 1, the resulting vector will have a smaller magnitude than the original. If the scalar is more than 1, the resulting vector will have a larger magnitude.