Euclidean norm

The Euclidean norm of a vector is the square root of the sum of its squared coordinates. It equals the vector’s ordinary geometric length in Euclidean space.

Dot product: An operation that multiplies corresponding vector coordinates and sums the products. Taking the dot product of a vector with itself and then its square root gives its Euclidean norm.
Vector: An object with magnitude and direction, represented in coordinates by an ordered list of numbers. The Euclidean norm assigns a length to each coordinate vector.
Manhattan norm: The norm that sums the absolute values of a vector’s coordinates. It measures axis-aligned travel, unlike the Euclidean norm’s straight-line length.
Euclidean distance: The straight-line distance between two points, computed as the norm of their difference. It uses the Euclidean norm to compare positions in coordinate space.
Pythagorean theorem: A theorem stating that the square of a right triangle’s hypotenuse equals the sum of the squares of its legs. The norm formula extends this theorem from two perpendicular components to any number of coordinates.
Euclidean space: A real vector space equipped with the standard inner product and its associated geometry. The norm is the standard length function on vectors in Euclidean space.
Maximum norm: The norm equal to the largest absolute coordinate of a vector. It measures the largest coordinate magnitude rather than combining all coordinates quadratically.
Least squares: A method that estimates unknowns by minimizing the sum of squared residuals. The residual sum of squares is the squared Euclidean norm of the error vector.
Triangle inequality: An inequality stating that the length of a sum is no greater than the sum of the lengths. The Euclidean norm satisfies this defining length bound for vector addition.
Absolute value: The nonnegative magnitude of a real number, equal to its distance from zero. In one dimension, the Euclidean norm reduces to the absolute value.

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