Euclidean metric (L2 norm) versus taxicab metric(L1 norm)

In summary, the Euclidean metric and taxicab metric are two different ways of measuring distance between two points. The Euclidean metric uses a straight line while the taxicab metric uses a grid-like path. The Euclidean metric is more commonly used in mathematical and geometric contexts while the taxicab metric is more commonly used in practical applications. The two metrics also differ in terms of distance calculation, with the Euclidean metric allowing for shorter distances. The Euclidean metric is more suitable for higher dimensions while the taxicab metric becomes less accurate. The two metrics cannot be used interchangeably and the appropriate one should be chosen based on the context and purpose of the measurement.
  • #1
mglaros
10
0

Homework Statement



I was just wondering how I would go about proving that the euclidean metric is always smaller than or equal to the taxicab metric for a given vector x in R^n. The result seems obvious but I am not sure how I would show this.

Homework Equations


The Attempt at a Solution

 
Physics news on Phys.org
  • #2
Show that (||x||1)2 - (||x||2)2 can never be negative.
 

1. What is the difference between Euclidean metric and taxicab metric?

The Euclidean metric, also known as the L2 norm, measures the distance between two points in a straight line. This is the most commonly used metric in geometry and is based on the Pythagorean theorem. On the other hand, the taxicab metric, also known as the L1 norm, measures the distance between two points by adding the absolute differences between their coordinates. This is often used in real-life situations where movement is restricted to a grid-like pattern, such as navigating through city blocks.

2. Which metric is more commonly used in real-world applications?

The answer to this question depends on the specific application. In general, the Euclidean metric is more commonly used in mathematical and geometric contexts, while the taxicab metric is more commonly used in practical applications, such as navigation and transportation.

3. How do the two metrics differ in terms of distance calculation?

The Euclidean metric calculates distance by finding the shortest straight line between two points, while the taxicab metric calculates distance by finding the shortest path along the edges of a grid-like structure. This means that the Euclidean metric can result in a shorter distance than the taxicab metric, as it allows for diagonal movement.

4. Which metric is more suitable for measuring distance in higher dimensions?

The Euclidean metric is more suitable for measuring distance in higher dimensions, as it takes into account all dimensions equally. The taxicab metric, on the other hand, becomes less accurate in higher dimensions as it only considers the differences between coordinates, rather than their absolute values.

5. Can the two metrics be used interchangeably?

No, the two metrics cannot be used interchangeably. They are fundamentally different ways of measuring distance and will result in different values. It is important to understand the context and purpose of the measurement in order to choose the appropriate metric to use.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
517
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Special and General Relativity
Replies
25
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Back
Top