Length (Magnitude, Distance)

Definition

Geometrically, the magnitude of a vector is a length of the directed line.

It is denoted by double vertical bars.

$$ ∣∣v∣∣ $$

The length(magnitude) of a vector quantifies its size(distance) from the origin in space.

Formula

It is computed using the Pythagorean theorem.

• 2D vector : $\vec{v}=(x,y)$
  • Length : $∣∣v∣∣=\sqrt{x^2+y^2}$
• 3D vector : $\vec{v}=(x,y,z)$
  • Length : $∣∣v∣∣=\sqrt{x^2+y^2+z^2}$
• n-Dimensional Vector : $\vec{v}=(x_1,x_2,\ldots,x_n)$
  • Length : $∣∣v∣∣=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}=\sqrt{\sum_{i=1}^n x_i^2}$

Properties

Non-Negativity

• The length of a vector is always non-negative.
  • $∣∣v∣∣≥0$
• If the length is zero, the vector is the zero vector
  • if $∣∣v∣∣=0$, then $\vec{v}=(0,0,0,…,0)$.

Triangle Inequality

$$ ∣∣u+v∣∣≤∣∣u∣∣+∣∣v∣∣ $$

• This property reflects the geometric fact that in any triangle.
• The length of one side cannot exceed the sum of the other two sides.