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Vector Cross Product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is denoted by the symbol \times. Given two linearly independent vectors and , the cross product, (read "a cross b"), is a vector that is perpendicular to both and , and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product). The magnitude of the cross product equals the area of a parallelogram with the vectors for sides; in particular, the magnitude of the product of two perpendicular vectors is the product of their lengths. The units of the cross-product are the product of the units of each vector. If two vectors are parallel or are anti-parallel (that is, they are linearly dependent), or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Product Vector
A cross is a religious symbol consisting of two Intersection (set theory), intersecting Line (geometry), lines, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is termed a saltire in heraldic terminology. The cross shape has been widely officially recognized as an absolute and exclusive religious symbol of Christianity from an early period in that religion's history.''Christianity: an introduction'' by Alister E. McGrath 2006 pages 321-323 Before then, it was used as a religious or cultural symbol throughout Europe, in West Asia, west and south Asia (the latter, in the form of the original Swastika); and in Ancient Egypt, where the Ankh was a hieroglyph that represented "life" and was used in the wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anticommutativity
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped arguments. The notion '' inverse'' refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of gives for example, Another prominent example of an anticommutative operation is the Lie bracket. In mathematical physics, where symmetry is of central importance, or even just in multilinear algebra these operations are mostly (multilinear with respect to some vector structures and then) called antisymmetric operations, and when they are not already of arity greater than two, extended in an associative setting to cover more than two arguments. Definition If A, B are two abelian groups, a bilinear map f\colon A^2 \to B is ant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate rotations in a space of any number of dimensions, and are a useful tool for classifying such rotations. Geometrically, a simple bivector can be interpreted as characterizing a directed plane segment (or oriented plane segment), much as vectors can be thought of as characterizing '' directed line segments''. The bivector has an ''attitude'' (or direction) of the plane spanned by and , has an area that is a scalar multiple of any reference plane segment with the same attitude (and in geometric algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector v in V. The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol \wedge and the fact that the product of two elements of V is "outside" V. The wedge product of k vectors v_1 \wedge v_2 \wedge \dots \wedge v_k is called a ''blade (geometry), blade of degree k'' or ''k-blade''. The wedge product was introduced originally as an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues: the magnitude (mathematics), magnitude of a bivector, -blade v\wedge w is the area of the parallelogram defined by v and w, and, more generally, the magnitude of a k-blade is the (hyper)volume of the Parallelepiped#Parallelotope, parallelotope defined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as rotations or translations, but which does ''not'' transform like a vector under certain ''discontinuous'' rigid transformations such as reflections. For example, the angular velocity of a rotating object is a pseudovector because, when the object is reflected in a mirror, the reflected image rotates in such a way so that ''its'' angular velocity "vector" is ''not'' the mirror image of the angular velocity "vector" of the ''original'' object; for true vectors (also known as ''polar vectors''), the reflection "vector" and the original "vector" ''must'' be mirror images. One example of a pseudovector is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b, [...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] |
Orientation (mathematics)
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimensiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
