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Rodrigues' Rotation Formula
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in , the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra to its Lie group . This formula is variously credited to Leonhard Euler, Olinde Rodrigues, or a combination of the two. A detailed historical analysis in 1989 concluded that the formula should be attributed to Euler, and recommended calling it "Euler's finite rotation formula." This proposal has received notable support, but some others have viewed the formula as just one of many variations of the Euler–Rodrigues formula, thereby crediting both. Statement If is a vector in and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Hand Rule
In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field. The various right- and left-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding your hands together with palms up and fingers curled. If the curl of the fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either right thumb or left thumb. History The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dimensions. William Rowan Hamilton, recognized for his development of quaternions, a mathematical system for rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular oriented lines, called '' coordinate lines'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the '' origin'' and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame. Similarly, the position of any point in three-dimensional space can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension . These coordinates are the signed distances from the point to mutually perpendicular fixed h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray (geometry), ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right-hand Rule
In mathematics and physics, the right-hand rule is a Convention (norm), convention and a mnemonic, utilized to define the orientation (vector space), orientation of Cartesian coordinate system, axes in three-dimensional space and to determine the direction of the cross product of two Euclidean vector, vectors, as well as to establish the direction of the force on a Electric current, current-carrying conductor in a magnetic field. The various right- and left-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding your hands together with palms up and fingers curled. If the curl of the fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either right thumb or left thumb. History The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Triple Product
In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. Scalar triple product The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation Geometrically, the scalar triple product : \mathbf\cdot(\mathbf\times \mathbf) is the (signed) volume of the parallelepiped defined by the three vectors given. Properties * The scalar triple product is unchanged under a circular shift of its three operands (a, b, c): *: \mathbf\cdot(\mathbf\times \mathbf)= \mathbf\cdot(\mathbf\times \mathbf)= \mathbf\cdot(\mathbf\times \mathbf) * Swapping the positions of the operators without re-ordering the operands lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Rejection
The vector projection (also known as the vector component or vector resolution) of a vector on (or onto) a nonzero vector is the orthogonal projection of onto a straight line parallel to . The projection of onto is often written as \operatorname_\mathbf \mathbf or . The vector component or vector resolute of perpendicular to , sometimes also called the vector rejection of ''from'' (denoted \operatorname_ \mathbf or ), is the orthogonal projection of onto the plane (or, in general, hyperplane) that is orthogonal to . Since both \operatorname_ \mathbf and \operatorname_ \mathbf are vectors, and their sum is equal to , the rejection of from is given by: \operatorname_ \mathbf = \mathbf - \operatorname_ \mathbf. To simplify notation, this article defines \mathbf_1 := \operatorname_ \mathbf and \mathbf_2 := \operatorname_ \mathbf. Thus, the vector \mathbf_1 is parallel to \mathbf, the vector \mathbf_2 is orthogonal to \mathbf, and \mathbf = \mathbf_1 + \mathbf_2. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Projection
The vector projection (also known as the vector component or vector resolution) of a vector on (or onto) a nonzero vector is the orthogonal projection of onto a straight line parallel to . The projection of onto is often written as \operatorname_\mathbf \mathbf or . The vector component or vector resolute of perpendicular to , sometimes also called the vector rejection of ''from'' (denoted \operatorname_ \mathbf or ), is the orthogonal projection of onto the plane (or, in general, hyperplane) that is orthogonal to . Since both \operatorname_ \mathbf and \operatorname_ \mathbf are vectors, and their sum is equal to , the rejection of from is given by: \operatorname_ \mathbf = \mathbf - \operatorname_ \mathbf. To simplify notation, this article defines \mathbf_1 := \operatorname_ \mathbf and \mathbf_2 := \operatorname_ \mathbf. Thus, the vector \mathbf_1 is parallel to \mathbf, the vector \mathbf_2 is orthogonal to \mathbf, and \mathbf = \mathbf_1 + \mathbf_2. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused with scalar multiplication. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two Euclidean vector, vectors is widely used. It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see ''Inner product space'' for more). It should not be confused with the cross product. Algebraically, the dot product is the sum of the Product (mathematics), products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Right Hand Rule
In mathematics and physics, the right-hand rule is a convention and a mnemonic, utilized to define the orientation of axes in three-dimensional space and to determine the direction of the cross product of two vectors, as well as to establish the direction of the force on a current-carrying conductor in a magnetic field. The various right- and left-hand rules arise from the fact that the three axes of three-dimensional space have two possible orientations. This can be seen by holding your hands together with palms up and fingers curled. If the curl of the fingers represents a movement from the first or x-axis to the second or y-axis, then the third or z-axis can point along either right thumb or left thumb. History The right-hand rule dates back to the 19th century when it was implemented as a way for identifying the positive direction of coordinate axes in three dimensions. William Rowan Hamilton, recognized for his development of quaternions, a mathematical system for rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Decomposition Unit Vector Rodrigues Rotation Formula
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically used for lines and planes that intersect to form a right angle, whereas ''orthogonal'' is used in generalizations, such as ''orthogonal vectors'' or ''orthogonal curves''. ''Orthogonality'' is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics Optics In optics, polarization states are said to be orthog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
