﻿

# Parallelogram law

In mathematics, the simplest form of the parallelogram law belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. With the notation in the diagram on the right, this can be stated as

: $AB^2+BC^2+CD^2+AD^2=AC^2+BD^2,.$

In case the parallelogram is a rectangle, the two diagonals are of equal lengths and the statement reduces to the Pythagorean theorem. But in general, the square of the length of "neither" diagonal is the sum of the squares of the lengths of two sides.

The parallelogram law in inner product spaces

In inner product spaces, the statement of the parallelogram law reduces to the algebraic identity

:$2|x|^2+2|y|^2=|x+y|^2+|x-y|^2$

where

:$|x|^2=langle x, x angle.$

Normed vector spaces satisfying the parallelogram law

Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (hence the name), and thus one can evaluate the expressions on both sides of "=" in the identity above. A remarkable fact is that if the identity above holds, then the norm must arise in the usual way from some inner product. Additionally, the inner product generating the norm is unique, as a consequence of the polarization identity; in the real case, it is given by

:$langle x, y angle=\left\{|x+y|^2-|x-y|^2over 4\right\},$

or, equivalently, by

:$\left\{|x+y|^2-|x|^2-|y|^2over 2\right\} ext\left\{ or \right\}\left\{|x|^2+|y|^2-|x-y|^2over 2\right\}.$

In the complex case it is given by

:$langle x, y angle=\left\{|x+y|^2-|x-y|^2over 4\right\}+i\left\{|ix-y|^2-|ix+y|^2over 4\right\}.$

* [http://www.unlvkappasigma.com/parallelogram_law/ The Parallelogram Law Proven Simply] at [http://www.unlvkappasigma.com/ UNLV Kappa Sigma]
* [http://www.cut-the-knot.org/Curriculum/Geometry/ParallelogramIdentity.shtml The Parallelogram Law: A Proof Without Words] at cut-the-knot
* [http://planetmath.org/?op=getobj&from=objects&name=ProofOfParallelogramLaw2 Proof of Parallelogram Law] at [http://planetmath.org/ Planet Math]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• parallelogram law — noun : a law in physics: the resultant of two vector quantities represented in magnitude, direction, and sense by two adjacent sides of a parallelogram both of which are directed toward or away from their point of intersection is the diagonal of… …   Useful english dictionary

• parallelogram law — Math., Physics. a rule for adding two vectors, as forces (parallelogram of forces), by placing the point of application of one at the point of origin of the other and obtaining their sum by constructing the line connecting the two remaining end… …   Universalium

• Law of cosines — This article is about the law of cosines in Euclidean geometry. For the cosine law of optics, see Lambert s cosine law. Figure 1 – A triangle. The angles α, β, and γ are respectively opposite the sides a, b, and c …   Wikipedia

• Pythagorean theorem — See also: Pythagorean trigonometric identity The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c) …   Wikipedia

• Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia

• Rhombus — For other uses, see Rhombus (disambiguation). Rhombus Two rhombi Type quadrilateral, bipyramid Edges and vertices …   Wikipedia

• Relative velocity — In kinematics, relative velocity is the vector difference between the velocities of two objects, as evaluated in terms of a single coordinate system, usually an inertial frame of reference unless specifically stated otherwise.For example, if the… …   Wikipedia

• Ley del paralelogramo — Un paralelogramo. Los lados de éste están mostrados en color azul y las diagonales en rojo. En matemática, la forma más simple de la ley del paralelogramo pertenece a la geometría elemental. Ésta postula que la suma de los cuadrados de las… …   Wikipedia Español

• vector analysis — the branch of calculus that deals with vectors and processes involving vectors. * * * ▪ mathematics Introduction       a branch of mathematics that deals with quantities that have both magnitude and direction. Some physical and geometric… …   Universalium