Vector Norms & Distance — Vector Norms and Distance |...

SECTION 1

Historical Context & Motivation

Have you ever wondered how your phone's GPS knows exactly how far you are from a restaurant? Or how a video game calculates whether an enemy is close enough to attack? These questions all come down to one idea: measuring distance. Mathematicians have spent centuries figuring out how to measure length and distance not just on a flat surface, but in spaces with many dimensions.

The story of vector norms (a fancy name for "the length of a vector") stretches back to ancient Greece and continues through modern computer science. Let's trace how this idea developed.

~300 BC

Euclid's Geometry

The Greek mathematician Euclid wrote Elements, which included the Pythagorean theorem — the foundation for measuring distance between two points on a flat surface.

1637

Descartes Introduces Coordinates

René Descartes created the coordinate system (the x-y plane you use in class), allowing distance to be calculated using algebra instead of just drawings.

1880s

Minkowski & Higher Dimensions

Hermann Minkowski generalized distance formulas beyond the familiar Euclidean version. He introduced the idea of different "norms" — different ways to measure length — which became essential in physics and math.

1922

Banach's Normed Spaces

Stefan Banach formally defined normed vector spaces, giving mathematicians a rigorous framework to study length and distance in abstract settings.

2000s+

Machine Learning & Data Science

Today, vector norms and distance are everywhere — from facial recognition to recommendation engines. Choosing the right norm can determine whether an algorithm works well or fails.

So here is the central question: How do we measure the "size" of a vector, and how do we use that measurement to find the distance between two points? That's exactly what this lesson will answer.

SECTION 2

Core Principles & Definitions

Before we dive into formulas, let's nail down some key ideas. A vector is a quantity that has both a direction and a magnitude (size). You can think of it as an arrow pointing from one spot to another. A norm is simply a rule that assigns a non-negative number to a vector — that number tells you the vector's length.

What Is a Norm?

A norm is a function that takes a vector and returns a single number representing its "length." It is always zero or positive.

Non-Negativity

A norm is never negative. The only vector with a norm of zero is the zero vector (the vector with all components equal to 0).

Scalar Multiplication

If you multiply a vector by a number c, the norm gets multiplied by |c|. Doubling a vector doubles its length.

Triangle Inequality

The norm of the sum of two vectors is always less than or equal to the sum of their individual norms. In plain English: a shortcut is never longer than going the long way around.

Distance = Norm of the Difference

The distance between two vectors u and v is defined as the norm of their difference: ‖u − v‖.

✦ KEY TAKEAWAY

Think of a norm like a ruler for arrows. Just as you can measure how long a pencil is with a ruler, a norm measures how long a vector is. And if you want to know how far apart two arrows' tips are, you subtract one vector from the other and then measure the result — that gives you the distance.

SECTION 3

Visual Explanation

Let's see what vector norms and distance look like on a coordinate plane. The diagram below shows two vectors, u = (3, 4) and v = (7, 1). Each vector's norm is its length from the origin, and the distance between them is the length of the dashed line connecting their tips.

The purple arrow is vector u = (3, 4) with norm 5. The cyan arrow is vector v = (7, 1) with norm ≈ 7.07. The dashed gold line shows the distance between their endpoints.

Notice that the length of the purple arrow (‖u‖ = 5) comes from the Pythagorean theorem: √(3² + 4²) = √(9 + 16) = √25 = 5. That's all a norm really is — a generalization of the Pythagorean theorem to measure vector length. The distance between u and v is the norm of the difference vector u − v = (3 − 7, 4 − 1) = (−4, 3), which gives ‖u − v‖ = √(16 + 9) = 5.

SECTION 4

Mathematical Framework

Now let's formalize the math. There are several different norms, but the most common ones you'll encounter are the L¹ norm (also called the Manhattan norm), the L² norm (also called the Euclidean norm), and the L∞ norm (also called the max norm). Each one measures "length" in a slightly different way.

EUCLIDEAN NORM (L² NORM)

‖v‖₂ = √(v₁² + v₂² + … + vₙ²)

This is the most familiar norm — it gives the straight-line distance from the origin to the tip of the vector. Here v₁, v₂, …, vₙ are the components of the vector v.

MANHATTAN NORM (L¹ NORM)

‖v‖₁ = |v₁| + |v₂| + … + |vₙ|

Add up the absolute values of all components. It's called the "Manhattan" norm because it measures distance the way you'd walk on a city grid — only horizontal and vertical moves, no cutting diagonally.

MAX NORM (L∞ NORM)

‖v‖∞ = max(|v₁|, |v₂|, …, |vₙ|)

Take the largest absolute value among all components. This norm only cares about the single biggest entry in the vector.

DISTANCE FORMULA

d(u, v) = ‖u − v‖

The distance between two vectors u and v is always the norm of their difference. Whichever norm you choose (L¹, L², or L∞), the distance formula stays the same — just plug in the appropriate norm.

SECTION 5

Comparing Different Norms

Different norms give different answers for the same vector. The diagram below shows the unit circles for the L¹, L², and L∞ norms. A "unit circle" for a norm is the set of all vectors whose norm equals exactly 1. Each norm defines a different shape.

The L¹ unit circle (diamond) is rotated 45°. The L² unit circle is the familiar round circle. The L∞ unit circle (square) captures all vectors whose largest component is at most 1.

Let's look at a specific example. For the vector v = (3, −4), the three norms give different values.

Comparing L¹, L², and L∞ norms for the vector (3, −4)

Norm	Formula	Calculation for v = (3, −4)	Result
L¹	\|v₁\| + \|v₂\|	\|3\| + \|−4\| = 3 + 4	7
L²	√(v₁² + v₂²)	√(9 + 16) = √25	5
L∞	max(\|v₁\|, \|v₂\|)	max(3, 4)	4

💡 Notice the Pattern

For any vector, the L∞ norm is always the smallest, the L¹ norm is always the largest, and the L² norm falls in between: ‖v‖∞ ≤ ‖v‖₂ ≤ ‖v‖₁. This makes sense when you look at the unit circle diagram — the square (L∞) contains the circle (L²), which contains the diamond (L¹).

SECTION 6

Worked Example

Let's work through a complete problem. Suppose you have two vectors in three-dimensional space: u = (1, −2, 3) and v = (4, 0, −1). Find the Euclidean (L²) distance between u and v.

Finding the Euclidean Distance Between Two 3D Vectors

Step 1 — Find the Difference VectorSubtract corresponding components: u − v = (1 − 4, −2 − 0, 3 − (−1)) = (1 − 4, −2 − 0, 3 + 1).

u − v = (−3, −2, 4)

Step 2 — Square Each ComponentSquare each entry of the difference vector: (−3)² = 9, (−2)² = 4, and 4² = 16.

Squares: 9, 4, 16

Step 3 — Sum the SquaresAdd up all the squared values: 9 + 4 + 16 = 29.

Sum = 29

Step 4 — Take the Square RootThe Euclidean distance is the square root of the sum: √29 ≈ 5.39.

d(u, v) = ‖u − v‖₂ = √29 ≈ 5.39

⚠️ Pro Tip

Always be careful with signs! When you subtract vectors, a double negative (like 3 − (−1)) becomes addition. This is one of the most common places students make mistakes.

SECTION 7

When to Use Each Norm

You might wonder: if the L² norm is the "natural" one, why bother with the others? It turns out that different situations call for different norms. Each norm captures a different idea of "closeness," and choosing the right one can make a big difference in real-world applications.

Comparing the three most common vector norms

Norm	Strengths	Limitations	Best Used For
L¹ (Manhattan)	Robust to outliers (extreme values). Easy to compute.	Doesn't give the shortest path geometrically.	City-block navigation, sparse data analysis, machine learning feature selection.
L² (Euclidean)	Matches physical intuition. Smooth and differentiable everywhere.	Sensitive to outliers due to squaring. Computationally heavier in high dimensions.	Physics calculations, GPS distance, k-nearest neighbor algorithms.
L∞ (Max)	Fastest to compute. Focuses on the worst-case component.	Ignores all components except the largest, losing information.	Error checking (worst-case analysis), chess-board distance, quality control.

✦ KEY TAKEAWAY

Choosing a norm is like choosing how to grade a test. The L¹ norm is like adding up all the point deductions — every mistake counts equally. The L² norm is like combining all errors into one "overall" score — big mistakes count more because of the squaring. The L∞ norm is like saying "your grade is determined by your worst single answer." Different classes might use different approaches, and each gives a different picture of performance.

SECTION 8

Connection to Advanced Theory

The concepts of norms and distance you've learned here are the foundation for much more advanced math. In college-level linear algebra, you'll encounter inner product spaces — places where the L² norm comes from a special operation called the inner product (also known as the dot product). The relationship is: ‖v‖₂ = √(v · v), where v · v is the dot product of the vector with itself.

From today's concepts to advanced linear algebra

What You Learned Today	Where It Leads
Norm of a vector (‖v‖)	Normed vector spaces, Banach spaces — abstract spaces where length is defined.
Euclidean distance d(u, v) = ‖u − v‖₂	Metric spaces — any space with a distance function obeying specific rules.
Different norms (L¹, L², L∞)	Lp norms for any p ≥ 1, function spaces, and optimization theory.
Triangle inequality	Cauchy-Schwarz inequality, orthogonality, and projection theorems.

Don't worry if those advanced terms sound intimidating right now! The important thing is that you already understand the core idea: norms measure length, and distance is the norm of the difference between two vectors. Every advanced topic in this area builds on exactly that foundation.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

In your own words, explain what a vector norm measures. Why must a norm always be non-negative?

PROBLEM 2 — BASIC CALCULATION

Find the Euclidean norm (L²) of the vector w = (5, −12).

PROBLEM 3 — INTERMEDIATE

Given u = (2, −1, 3) and v = (−1, 4, 1), find the distance between u and v using the L¹ (Manhattan) norm.

PROBLEM 4 — APPLIED

A drone is at position (10, 20, 50) in meters, and its landing pad is at (30, 5, 0). Find the straight-line distance (Euclidean distance) between the drone and the pad. Then find the L∞ distance and explain what it represents physically.

PROBLEM 5 — CRITICAL THINKING

Prove that the L∞ norm of any vector in ℝⁿ is always less than or equal to its L¹ norm. That is, show ‖v‖∞ ≤ ‖v‖₁ for any vector v. Use the definitions of both norms in your argument.

SUMMARY

Lesson Summary

A vector norm is a function that assigns a non-negative number to a vector, representing its length. The three most important norms are the L¹ (Manhattan) norm, which sums the absolute values of all components; the L² (Euclidean) norm, which uses the Pythagorean theorem to find straight-line length; and the L∞ (max) norm, which returns the largest absolute component. Every valid norm satisfies three properties: non-negativity, scalar multiplication, and the triangle inequality.

The distance between two vectors is always defined as the norm of their difference: d(u, v) = ‖u − v‖. Different norms produce different distance values for the same pair of vectors, and the choice of norm depends on the application. The unit circle for each norm has a distinct shape — a diamond for L¹, a circle for L², and a square for L∞ — which visually captures how each norm "sees" the world. These ideas form the backbone of inner product spaces and orthogonality, two of the most powerful concepts in advanced linear algebra.