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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.
In Cartesian coordinates, for = + + the curl is the vector ... The generalization of the dot product formula to Riemannian manifolds is a defining property of a ...
The formula for the closest point to the origin may be expressed more succinctly using notation from linear algebra. The expression a x + b y + c z {\displaystyle ax+by+cz} in the definition of a plane is a dot product ( a , b , c ) ⋅ ( x , y , z ) {\displaystyle (a,b,c)\cdot (x,y,z)} , and the expression a 2 + b 2 + c 2 {\displaystyle a^{2 ...
The formula for calculating it can be derived and expressed in several ways. Knowing the shortest distance from a point to a line can be useful in various situations—for example, finding the shortest distance to reach a road, quantifying the scatter on a graph, etc.
The dot product in Cartesian coordinates (Euclidean space with an orthonormal basis set) is simply the sum of the products of components. In orthogonal coordinates, the dot product of two vectors x and y takes this familiar form when the components of the vectors are calculated in the normalized basis:
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set R 2 {\displaystyle \mathbb {R} ^{2}} 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.
This is known as triple product expansion, or Lagrange's formula, [2] [3] ... The dot product of two vectors is a scalar but the dot product of a pseudovector and a ...
The dot product takes two vectors x and y, and produces a real number x ⋅ y. If x and y are represented in Cartesian coordinates, then the dot product is defined by () = + +. The dot product satisfies the properties [1] It is symmetric in x and y: x ⋅ y = y ⋅ x.