Search results
Results from the WOW.Com Content Network
If the cross product of two vectors is the zero vector (that is, a × b = 0), then either one or both of the inputs is the zero vector, (a = 0 or b = 0) or else they are parallel or antiparallel (a ∥ b) so that the sine of the angle between them is zero (θ = 0° or θ = 180° and sin θ = 0). The self cross product of a vector is the zero ...
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.
Including 0, the set has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice of base giving a logarithmic unit) gives an isomorphism with the log semiring (with 0 corresponding to ), and its units (the finite numbers, excluding ) correspond to the positive real numbers.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
Zero to the power of zero, denoted as 0 0, is a mathematical expression with different interpretations depending on the context. In certain areas of mathematics, such as combinatorics and algebra, 0 0 is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents.
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.
The notation > means that the matrix is positive definite. Theorem (continuous time version). Given any Q > 0 {\displaystyle Q>0} , there exists a unique P > 0 {\displaystyle P>0} satisfying A T P + P A + Q = 0 {\displaystyle A^{T}P+PA+Q=0} if and only if the linear system x ˙ = A x {\displaystyle {\dot {x}}=Ax} is globally asymptotically stable.
Wiener measure on the space of continuous paths in is a strictly positive measure — Wiener measure is an example of a Gaussian measure on an infinite-dimensional space. Lebesgue measure on R n {\displaystyle \mathbb {R} ^{n}} (with its Borel topology and σ {\displaystyle \sigma } -algebra) is strictly positive.