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.