Lists Of Vector Identities (other)
There are two lists of mathematical identities related to vectors: * Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. *Vector calculus identities The following are important identities involving derivatives and integrals in vector calculus. Operator notation Gradient For a function f(x, y, z) in three-dimensional Cartesian coordinate variables, the gradient is the vector field: : ...

Vector Algebra Relations
The following are important identities in vector algebra. Identities that only involve the magnitude of a vector \, \mathbf A\, 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. Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier. Magnitudes The magnitude of a vector A can be expressed using the dot product: :\, \mathbf A \, ^2 = \mathbf In three-dimensional Euclidean space, the magnitude of a vector is determined from its three components using Pythagoras' theorem: :\, \mathbf A \, ^2 = A_1^2 + A_2^2 +A_3^2 Inequalities *The Cauchy–Schwarz inequality: \mathbf \cdot \mathbf \le \left\, \mathbf A \right\, \left\, \mathbf B \right\, *The triangle inequality: \, \mathbf\, \le \, \mathbf\, + \, \mathbf\, *The reverse triangle inequ ...
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