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For two elements a 1 + b 1 i + c 1 j + d 1 k and a 2 + b 2 i + c 2 j + d 2 k, their product, called the Hamilton product (a 1 + b 1 i + c 1 j + d 1 k) (a 2 + b 2 i + c 2 j + d 2 k), is determined by the products of the basis elements and the distributive law. The distributive law makes it possible to expand the product so that it is a sum of ...
Euclidean vectors such as (2, 3, 4) or (a x, a y, a z) can be rewritten as 2 i + 3 j + 4 k or a x i + a y j + a z k, where i, j, k are unit vectors representing the three Cartesian axes (traditionally x, y, z), and also obey the multiplication rules of the fundamental quaternion units by interpreting the Euclidean vector (a x, a y, a z) as the ...
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs.
By referring collectively to e 1, e 2, e 3 as the e basis and to n 1, n 2, n 3 as the n basis, the matrix containing all the c jk is known as the "transformation matrix from e to n", or the "rotation matrix from e to n" (because it can be imagined as the "rotation" of a vector from one basis to another), or the "direction cosine matrix from e ...
A system of two angular momenta with magnitudes j 1 and j 2 can be described either in terms of the uncoupled basis states (labeled by the quantum numbers m 1 and m 2), or the coupled basis states (labeled by j 3 and m 3). The 3-j symbols constitute a unitary transformation between these two bases, and this unitarity implies the orthogonality ...
The notations (î, ĵ, k̂), (x̂ 1, x̂ 2, x̂ 3), (ê x, ê y, ê z), or (ê 1, ê 2, ê 3), with or without hat, are also used, [1] particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with index symbols such as i, j, k, which are used to identify an element of a set or array or sequence of ...
Conversely, if k 2 is less than one, F 2 −F 1 is a positive number; the added inverse-cube force is repulsive. If k is an integer such as 3, the orbit of the second particle is said to be a harmonic of the first particle's orbit; by contrast, if k is the inverse of an integer, such as 1 ⁄ 3, the second orbit is said to be a subharmonic of ...