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The line segments AB and CD are orthogonal to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Whereas perpendicular is typically followed by to when relating two lines to one another (e.g., "line A is perpendicular to line B"), [1] orthogonal is commonly used without to (e.g., "orthogonal lines A and B").
Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions sin nx and sin mx are orthogonal on the interval x ∈ ( − π , π ) {\displaystyle x\in (-\pi ,\pi )} when m ≠ n {\displaystyle m\neq n} and n and m are positive integers.
An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 0-677-04150-0. Chihara, Theodore Seio (2001). "45 years of orthogonal polynomials: a view from the wings". Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999).
Example of orthogonal factorial design Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out. Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal.
The Staudinger ligation is a reaction developed by the Bertozzi group in 2000 that is based on the classic Staudinger reaction of azides with triarylphosphines. [15] It launched the field of bioorthogonal chemistry as the first reaction with completely abiotic functional groups although it is no longer as widely used.
In evolutionary biology, function is the reason some object or process occurred in a system that evolved through natural selection. That reason is typically that it achieves some result, such as that chlorophyll helps to capture the energy of sunlight in photosynthesis .
Orthogonal projection onto a line, m, is a linear operator on the plane. This is an example of an endomorphism that is not an automorphism. In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism.
This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces.