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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.
The inner product of a Euclidean space is often called dot product and denoted x ⋅ y. This is specially the case when a Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate vectors. For this reason, and for historical reasons, the dot notation is more commonly ...
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane, since every Euclidean plane is isomorphic to it.
The formula for the closest point to the origin may be expressed more succinctly using notation from linear algebra.The expression + + in the definition of a plane is a dot product (,,) (,,), and the expression + + appearing in the solution is the squared norm | (,,) |.
The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted ‖ x ‖, and to the angle θ between two vectors x and y by means of the formula = ‖ ‖ ‖ ‖ .
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set of the ordered pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane or standard Euclidean plane, since every Euclidean plane is isomorphic to it.
The dot product of two vectors tangent to the sphere sitting inside 3-dimensional Euclidean space contains information about the lengths and angle between the vectors. The dot products on every tangent plane, packaged together into one mathematical object, are a Riemannian metric.
The usual Euclidean dot product in ℝ m is a metric which, when restricted to vectors tangent to M, gives a means for taking the dot product of these tangent vectors. This is called the induced metric. Suppose that v is a tangent vector at a point of U, say