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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.
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 ...
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.
The first three operations below assume that x = b c and/or y = b d ... There are multiple ways to derive/prove each logarithm law – this is just one possible ...
It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution, if and only if A and B are invertible (Horn & Johnson 1991, Lemma 4.3.1). If X and C are row-ordered into the column vectors u and v , respectively, then ( Jain 1989 , 2.8 Block Matrices and Kronecker Products)
The multiplicative identity of R[x] is the polynomial x 0; that is, x 0 times any polynomial p(x) is just p(x). [2] Also, polynomials can be evaluated by specializing x to a real number. More precisely, for any given real number r, there is a unique unital R-algebra homomorphism ev r : R[x] → R such that ev r (x) = r. Because ev r is unital ...
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
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.