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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").
We say that functions and are orthogonal if their inner product (equivalently, the value of this integral) is zero: f , g w = 0. {\displaystyle \langle f,g\rangle _{w}=0.} Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to another inner product.
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.
However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, the real functions span the same space as the complex ones would.
For example, the three-dimensional Cartesian coordinates (x, y, z) is an orthogonal coordinate system, since its coordinate surfaces x = constant, y = constant, and z = constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates.
These combinations are chosen to satisfy two conditions. First, the total amount of s and p orbital contributions must be equivalent before and after hybridisation. Second, the hybrid orbitals must be orthogonal to each other. [27] [28] If two hybrid orbitals were not orthogonal, by definition they would have nonzero orbital overlap. Electrons ...
A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T (v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between ...
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.