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In mathematics, the dot product or scalar product [note 1] 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 vectors is widely used.
Geometrically, and are parallel if their geometric product is equal to their inner product, whereas and are perpendicular if their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with the dot product of ...
A dot product representation of a simple graph is a method of representing a graph using vector spaces and the dot product from linear algebra. Every graph has a dot ...
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.
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
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
The dot product takes in two vectors and returns a scalar, while the cross product [a] returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics , and engineering .
2.6 Dot product rule. 2.7 Cross product rule. 3 Second derivative identities. ... When the Laplacian is equal to 0, the function is called a harmonic function.