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The dot product of two unit vectors behaves just oppositely: it is zero when the unit vectors are perpendicular and 1 if the unit vectors are parallel. Unit vectors enable two convenient identities: the dot product of two unit vectors yields the cosine (which may be positive or negative) of the angle between the two unit vectors.
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.
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.
The cross product of two vectors u and v would be represented as: By some conventions (e.g. in France and in some areas of higher mathematics), this is also denoted by a wedge, [ 12 ] which avoids confusion with the wedge product since the two are functionally equivalent in three dimensions: u ∧ v {\displaystyle \mathbf {u} \wedge \mathbf {v} }
In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero.
Cartesian product of the sets {x,y,z} and {1,2,3}In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. [1]
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.
In Euclidean 3-space, the wedge product has the same magnitude as the cross product (the area of the parallelogram formed by sides and ) but generalizes to arbitrary affine spaces and products between more than two vectors. Tensor product – for two vectors and , where and are vector spaces, their tensor product belongs to the tensor product ...