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The Euclidean distance between two points P = (x, y, z) and Q = (a, b, c) in space is defined as d(P, Q) = p(x − a)2 + (y − b)2 + (z − c)2. This is a definition not a result! It is motivated by Pythagoras theorem, but we will prove the later.
The Euclidean space R appears in data science as nreal data points can be seen as a vector. The Euclidean distance between two data points x= (x 1,...,x n) and a= (a 1,...,a n) is then d(x,a)2 = P n k=1 (x k−a k) 2. The sum of the squares appears in statistics, like in least square problems.
The zero 0 divides the positive axis from the negative axis. A point P in the plane R2 is determined by two coordinates. We write P = (x; y). In space R3 nally, we require three coordinates P = (x; y; z), where z usually is thought of as height, the distance from the xy-plane.
The Euclidean distance between two points P = (x,y,z) and Q = (a,b,c) in space is defined as d(P,Q) = q (x −a)2 +(y − b)2 +(z −c)2. This Euclidean distance is a definition but motivated by Pythagoras theorem. 4 Problem: Find the distance d(P,Q) between the points P = (1,2,5) and Q = (−3,4,7) and verify that√ d(P,M) + d(Q,M) = d(P,Q).
De nition: The Euclidean distance between two vectors x and exin Rn is the length of the vector ex x, i.e., kxe xk. Another useful concept in Euclidean space is the dot product, which is closely linked to the concept
simplest way of measuring distance between two destinations. However, this distance can be calculated using different formulas based on a variety of assumptions. Pythagorean Theorem or Euclidean Distance Distance = sqrt((X2 - X1)^2 + (Y2 - Y1)^2) where X1, Y1 and X2, Y2 are the Cartesian coordinates of your destinations.
Euclidean distance. The immediate consequence of this is that the squared length of a vector x = [ x1 x2 ] is the sum of the squares of its coordinates (see triangle OPA in Exhibit 4.2, or triangle OPB – . | denotes the squared length of x, that is the distance between point O and P); and the .
