In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the...
9 KB (1,630 words) - 20:56, 31 August 2024
a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are...
15 KB (1,995 words) - 03:18, 3 October 2024
Rhombus (redirect from Equilateral parallelogram)
of the sides equals the sum of the squares of the diagonals (the parallelogram law). Thus denoting the common side as a and the diagonals as p and q...
16 KB (1,719 words) - 20:12, 15 September 2024
p^{2}+q^{2}=2(m^{2}+n^{2}).} This is also a corollary to the parallelogram law applied in the Varignon parallelogram. The lengths of the bimedians can also be expressed...
48 KB (6,876 words) - 11:23, 23 September 2024
product. The norm associated with any inner product space satisfies the parallelogram law: ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 . {\displaystyle...
22 KB (3,821 words) - 18:17, 1 June 2024
Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection...
128 KB (17,488 words) - 18:46, 10 October 2024
rest d0 ones are degenerate and form the U ∩ U⊥ intersection. The parallelogram law takes the form q ( x ) + q ( y ) = 1 2 ( q ( x + y ) + q ( x − y )...
19 KB (2,367 words) - 07:09, 14 July 2024
Varignon's theorem (redirect from Varignon parallelogram)
of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was...
10 KB (1,184 words) - 12:02, 7 August 2024
Pythagorean theorem (redirect from Pythagoras's Law)
theorem in an inner product space to non-orthogonal vectors is the parallelogram law: 2 ‖ v ‖ 2 + 2 ‖ w ‖ 2 = ‖ v + w ‖ 2 + ‖ v − w ‖ 2 , {\displaystyle...
92 KB (12,566 words) - 06:51, 6 September 2024
{\displaystyle x} and y {\displaystyle y} are linearly dependent. Parallelogram law ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 {\displaystyle...
56 KB (7,305 words) - 10:53, 26 August 2024
convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem...
5 KB (808 words) - 21:01, 30 June 2021
distance through the Pythagorean theorem and its generalization, the parallelogram law. Euclidean distance is not a smooth function: the three-dimensional...
15 KB (1,988 words) - 07:48, 23 August 2024
the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law. The theorem can be proved as a special...
3 KB (478 words) - 15:28, 19 May 2024
lengths of the tangent vectors fulfil the axioms of a norm and the parallelogram law, then the length of a vector is the square root of the value of the...
17 KB (2,760 words) - 00:06, 19 July 2024
does not arise from any inner product, as it does not satisfy the parallelogram law. It is not entirely straightforward to construct natural examples...
33 KB (6,387 words) - 21:50, 28 September 2024
Commutative property (redirect from Commutative law)
commutant) Commutative diagram Commutative (neurophysiology) Commutator Parallelogram law Particle statistics (for commutativity in physics) Proof that Peano's...
19 KB (2,203 words) - 01:59, 30 September 2024
The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object. When more than two forces are involved...
10 KB (1,599 words) - 06:20, 13 September 2024
+i\left(\|x+iy\|^{2}-\|x-iy\|^{2}\right)\right).} To see that the parallelogram law is sufficient, one observes in the real case that ⟨ x , y ⟩ {\displaystyle...
104 KB (17,224 words) - 06:29, 3 October 2024
British flag theorem for rectangles equates two sums of two squares The parallelogram law equates the sum of the squares of the four sides to the sum of the...
4 KB (704 words) - 22:13, 18 November 2023
generally, a polytope (in arbitrary finite dimension). The parallelogram law (called also parallelogram identity) ‖ x − y ‖ 2 + ‖ x + y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖...
69 KB (9,326 words) - 21:09, 27 August 2024
of a triangle Euler's quadrilateral theorem, an extension of the parallelogram law to convex quadrilaterals Euler–Rodrigues formula concerning Euler–Rodrigues...
14 KB (1,605 words) - 11:26, 7 October 2024
and c, respectively; if γ is acute, then ab cos γ is the area of the parallelogram with sides a and b forming an angle of γ′ = π/2 − γ; if γ is obtuse...
36 KB (5,707 words) - 22:16, 18 August 2024
p-norm is not an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of p is called Minkowski's...
34 KB (5,176 words) - 13:40, 1 July 2024
there to exist such an inner product is for the norm to satisfy the parallelogram law: ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 for all...
11 KB (1,434 words) - 18:52, 9 November 2023
convexity of F. Same for strict convexity. Linearity in F, law of cosines, parallelogram law: by definition. Duality: See figure 1 of. Bregman balls are...
26 KB (4,467 words) - 00:18, 30 September 2024
{\displaystyle f(x+y)+f(x-y)=2[f(x)+f(y)]\,\!} (quadratic equation or parallelogram law) f ( ( x + y ) / 2 ) = ( f ( x ) + f ( y ) ) / 2 {\displaystyle...
12 KB (1,632 words) - 11:28, 19 August 2024
since any norm that is induced by an inner product should satisfy the parallelogram law ‖ x + y ‖ p 2 + ‖ x − y ‖ p 2 = 2 ‖ x ‖ p 2 + 2 ‖ y ‖ p 2 . {\displaystyle...
22 KB (3,603 words) - 16:12, 23 February 2024
tessellation Voronoi tessellation Delaunay triangulation Quasicrystal Parallelogram law Polytope Schläfli symbol Regular polytope Regular Polytopes Sphere...
13 KB (910 words) - 13:17, 13 September 2024
on X ¯ ′ {\displaystyle {\overline {X}}^{\prime }} satisfies the parallelogram law, which means that the polarization identity can be used to define...
10 KB (1,769 words) - 15:02, 31 August 2024
through this. Since the cross product of two vectors gives the area of a parallelogram possessing sides of those vectors, the triangular area dA swept out...
60 KB (8,823 words) - 11:35, 2 October 2024