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"Scalar product" redirects here. For the abstract scalar product, see Inner product space. For the operation on complex vector spaces, see Hermitian form. For the product of a vector and a scalar, see Scalar multiplication.
In mathematics, the dot product, or scalar product (or sometimes inner product in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the magnitudes of the two vectors and the cosine of the angle between them. The name "dot product" is derived from the centered dot " " that is often used to designate this operation; the alternative name "scalar product" emphasizes the scalar (rather than vectorial) nature of the result.
In three dimensional space, the dot product contrasts with the cross product of two vectors, which produces a pseudovector as result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions. source wikipedia
The dot product is often defined in one of two ways: algebraically or geometrically. Equivalence of these definitions is proven later.
The dot product of two vectors a = [a1, a2, ..., an] and b = [b1, b2, ..., bn] is defined as:
where Σ denotes summation notation and n is the dimension of the vector space.
In two-dimensional space, the dot product of vectors [a, b] and [c, d] is ac + bd.
Similarly, in a three-dimensional space, the dot product of vectors [a, b, c] and [d, e, f] is ad + be + cf. For example, if [1, 3, −5] and [4, −2, −1] their dot product is:
Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix.
In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by . The dot product of two Euclidean vectors A and B is defined by
where θ is the angle between A and B.
In particular, if A and B are orthogonal, then the angle between them is 90°, so in that case
At the other extreme, if B = A, then the angle is 0°, and the dot product is just the length of A squared:
Scalar projection an
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