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In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their ...
Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
In mathematics, vector multiplication may refer to one of several operations between two (or more) vectors. It may concern any of the following articles: Dot product – also known as the "scalar product", a binary operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defined as the product of the ...
The vector triple product is defined as the cross product of one vector with the cross product of the other two. The following relationship holds: The following relationship holds: a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c {\displaystyle \mathbf {a} \times (\mathbf {b} \times \mathbf {c} )=(\mathbf {a} \cdot \mathbf {c} )\mathbf {b ...
A general 3d rotation of a vector a, about an axis in the direction of a unit vector ω and anticlockwise through angle θ, can be performed using Rodrigues' rotation formula in the dyadic form a r o t = R ⋅ a , {\displaystyle \mathbf {a} _{\mathrm {rot} }=\mathbf {R} \cdot \mathbf {a} \,,}
where the operator denotes a dot product, ^ is the unit vector in the direction of , ‖ ‖ is the length of , and is the angle between and . [ 1 ] The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates , the components of a vector are the scalar projections in the directions of the coordinate axes .
In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions n and m , then their outer product is an n × m matrix.
The most noteworthy property of cosine similarity is that it reflects a relative, rather than absolute, comparison of the individual vector dimensions. For any positive constant and vector , the vectors and are maximally similar. The measure is thus most appropriate for data where frequency is more important than absolute values; notably, term ...