EUCLIDEAN NORM Meaning and
Definition
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The Euclidean norm, also known as the Euclidean length or 2-norm, is a measure of the magnitude or length of a vector in a Euclidean space. It is a fundamental concept in linear algebra and mathematics. Specifically, it is a way of quantifying the distance between two points or the magnitude of a vector in n-dimensional space.
The Euclidean norm of a vector v = (v₁, v₂, ..., vₙ) in n-dimensional space is calculated as the square root of the sum of the squares of its components. In other words, it is the square root of (v₁² + v₂² + ... + vₙ²). This computation derives from the Pythagorean theorem: the length of the hypotenuse of a right-angled triangle can be found by taking the square root of the sum of the squares of the other two sides.
The Euclidean norm satisfies several important properties. It is always non-negative and equals zero if and only if the vector is the zero vector. The norm also obeys the triangle inequality, meaning that for any two vectors u and v, the norm of u plus v is less than or equal to the sum of the norms of u and v.
The Euclidean norm is widely used in various fields of science, engineering, and computer programming, such as physics, signal processing, machine learning, and optimization. It provides a concise representation of the magnitude of a vector and is often employed to compare and measure the similarity or dissimilarity between vectors in different applications.
Common Misspellings for EUCLIDEAN NORM
- wuclidean norm
- suclidean norm
- duclidean norm
- ruclidean norm
- 4uclidean norm
- 3uclidean norm
- eyclidean norm
- ehclidean norm
- ejclidean norm
- eiclidean norm
- e8clidean norm
- e7clidean norm
- euxlidean norm
- euvlidean norm
- euflidean norm
- eudlidean norm
- euckidean norm
- eucpidean norm
- eucoidean norm
- eucludean norm
Etymology of EUCLIDEAN NORM
The term "Euclidean norm" is derived from Euclidean geometry, named after the ancient Greek mathematician Euclid of Alexandria. Euclid is famous for his work "Elements", a comprehensive mathematical treatise that laid the foundation for Euclidean geometry and became one of the most influential works in the field. The Euclidean norm is a mathematical concept related to measuring distances in Euclidean space, which is the geometric space described by Euclid's axioms. The norm is often referred to as the "Euclidean norm" to emphasize its association with this geometric framework.
