Vector Norm Calculator

How to Calculate Vector Norms

Vector norms are measures of the "length" or "magnitude" of vectors. They are essential in various fields of mathematics, physics, and engineering. This guide will explain how to calculate the three most common vector norms: L1, L2, and L∞.

Vector Norm Formulas

For a vector \(\vec{v} = (v_1, v_2, \ldots, v_n)\), the norms are defined as follows:

• L1 Norm (Manhattan Norm): \(\|\vec{v}\|_1 = \sum_{i=1}^n |v_i|\)
• L2 Norm (Euclidean Norm): \(\|\vec{v}\|_2 = \sqrt{\sum_{i=1}^n v_i^2}\)
• L∞ Norm (Chebyshev Norm): \(\|\vec{v}\|_\infty = \max_{i} |v_i|\)

Calculation Steps

1. Identify the components of the vector.
2. For L1 Norm:
   • Take the absolute value of each component.
   • Sum all the absolute values.
3. For L2 Norm:
   • Square each component.
   • Sum the squared values.
   • Take the square root of the sum.
4. For L∞ Norm:
   • Take the absolute value of each component.
   • Find the maximum of these absolute values.

Example Calculation

Let's calculate the norms for the vector \(\vec{v} = (3, -4, 5)\):

L1 Norm:

\(\|\vec{v}\|_1 = |3| + |-4| + |5| = 3 + 4 + 5 = 12\)

L2 Norm:

\(\|\vec{v}\|_2 = \sqrt{3^2 + (-4)^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.071\)

L∞ Norm:

\(\|\vec{v}\|_\infty = \max(|3|, |-4|, |5|) = 5\)

Visual Representation

This diagram illustrates the different norms for a 2D vector. The blue arrow represents the vector, the red path shows the L1 norm (sum of absolute coordinates), the green line represents the L2 norm (Euclidean distance), and the purple square represents the L∞ norm (maximum absolute coordinate).