Vector Norms (original) (raw)

Last Updated : 23 Jul, 2025

A **vector norm, sometimes represented with a double bar as **∥x∥, is a function that assigns a non-negative length or size to a vector **x in n-dimensional space. Norms are essential in mathematics and machine learning for measuring vector magnitudes and calculating distances.

A vector norm satisfies three properties:

Types of Vector Norms

The vector norm **∣x∣ p, also known as the **p-norm, for p = 1, 2,… is defined as:

| \mathbf{x} |_p = \left( \sum_{i=1}^{n} | x_i |^p \right)^{\frac{1}{p}}

This general formula encompasses several specific norms that are commonly used.

Commonly used norms are:

Let's discuss these in detail.

**L 1 Norm

The **L 1 norm, also known as the **Manhattan norm or **Taxicab norm, is a way to measure the "length" or "magnitude" of a vector by summing the absolute values of its components.

Mathematically, for a vector x = [x1, x2, . . ., xn], the L1 norm ∣x∣1​ is defined as:

∣x∣1 = ∣x1∣ + ∣x2∣ + ∣x3∣ + ... + ∣xn∣

**Example: If x = [3, −4, 2], then the L1 norm is:

∣x∣1 ​= ∣3∣ + ∣−4∣ + ∣2∣ = 3 + 4 + 2 = 9

**L 2 Norm

The **L 2 norm, also known as the **Euclidean norm, is a measure of the "length" or "magnitude" of a vector, calculated as the square root of the sum of the squares of its components.

For a vector x = [x1, x2, . . ., xn], the L2 norm ∣x∣2​ is defined as:

| \mathbf{x} |_2 = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}

**Example: If x = [3, −4, 2], then the L2 norm is:

| \mathbf{x} |_2 = \sqrt{3^2 + (-4)^2 + 2^2}
= \sqrt{9 + 16 + 4}
=√29 ≈ 5.39

**L∞ norm

The **L∞ norm, also known as the **Infinity norm or **Max norm, measures the "size" of a vector by taking the largest absolute value among its components. Unlike the L1 and L2 norms, which consider the combined contribution of all components, the L∞ norm focuses solely on the component with the maximum magnitude.

For a vector x = [x1, x2, . . ., xn], the L∞ norm ∣x∣∞​ is defined as:

∣x∣∞ ​= max​∣xi​∣ where 1 ≤ i ≤ n

Example: If x = [3, −4, 2], then the L∞ norm is:

∣x∣∞​= max(∣3∣, ∣−4∣, ∣2∣) = 4

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Practice Problem Based on Vector Norm

**Question 1. Given the vector x = [4, -3, 7, 1], calculate the L1 norm (Manhattan norm) of the vector.
**Question 2. Given the vector x = [1, -2, 2], calculate the L2 norm (Euclidean norm) of the vector.
**Question 3. For the vector x = [7, −1, −4, 6], calculate the L∞ norm (Infinity norm) of the vector.
**Question 4. If the L2 norm (Euclidean norm) of a vector x = [x1, x2, x3] is 10, and the components of the vector are x1 = 6 and x2 = 8, find the value of x3.

**Answer:-

**1. 15
**2. 3
**3. 7
**4. 0