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Encyclopedia > 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. 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... A parallelogram. ... In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...


The parallelogram law in inner product spaces

In inner product spaces, the statement of the parallelogram law reduces to the algebraic identity In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

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

where

|x|^2=langle x, xrangle.

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 the identity above holds only if the norm is one that arises in the usual way from an 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 In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... A vector space whose norm is defined in terms of its inner product satisfies the polarization identity: This identity is analogous to the formula for the square of a binomial: If y in equation (1) is replaced by −y the result is which corresponds to the cosine law and...

langle x, yrangle={|x+y|^2-|x-y|^2over 4},

or, equivalently, by

|x+y|^2-|x|^2-|y|^2over 2 or {|x|^2+|y|^2-|x-y|^2over 2}.

In the complex case it is given by

langle x, yrangle={|x+y|^2-|x-y|^2over 4}+i{|ix-y|^2-|ix+y|^2over 4}.

External link

  • The Parallelogram Law: A Proof Without Words at cut-the-knot

  Results from FactBites:
 
Parallelogram - Wikipedia, the free encyclopedia (271 words)
Every parallelogram is a polygon, and more specifically a quadrilateral.
The three-dimensional counterpart of a parallelogram is a parallelepiped.
The area of a parallelogram can be seen as twice the area of a triangle created by one of its diagonals.
  More results at FactBites »

 
 

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