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The dot product of two vectors = ... is the unit vector in the direction of . Distributive law for the dot product. The dot product is thus ...
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. The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A ...
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 — regarding operations on vector fields such as divergence, gradient, curl, etc.
Using the modern terms cross product (×) and dot product (.), the quaternion product of two vectors p and q can be written pq = –p.q + p×q. In 1878, W. K. Clifford severed the two products to make the quaternion operation useful for students in his textbook Elements of Dynamic.
The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector.
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.
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in ^ (pronounced "v-hat"). The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,
In addition, the notion of direction is strictly associated with the notion of an angle between two vectors. If the dot product of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives a convenient algebraic characterization of both angle (a function of the dot ...