Pythagorean distance

The polar coordinate system describes the location of a point (denoted as [r,0]) in a plane by specifying a distance r and an angle 0 from the origin of the system. There are several relationships between polar and rectangular coordinates, diagrammed in Figure 1-30. From the Pythagorean Theorem... 

The squared Euclidean distance (also called Pythagorean distance) has been defined in Section 9.2.3 ... 

Figure 11-1 The distance between two points in a two-dimensional coordinate space is determined using the Pythagorean theorem.

The distances that Shirley and Shelly walked are the a and b of a right triangle. The distance that they re apart, 17 miles, is the c value. Substituting into the Pythagorean theorem, the equation a2 + b2 = c2 becomes (y)2 + (2y - l)2 =172. 

In the case of Shirley and Shelly who walked away from each other in directions that are right angles from one another, you find that Shirley walked 8 miles and Shelly walked 15 miles. They re only 17 miles apart — even though they walked a total of 23 miles — because their journeys and the distance between them formed a right triangle. They didn t walk in opposite directions. As far as the accuracy of the solution, check out the Pythagorean theorem with the distances ... 

What if two jets leave at different times, and you want to determine the speed at which a particular jet is traveling Use the distance formula and the Pythagorean theorem. 

Find the two distances along the hypotenuses (in terms of x) using the Pythagorean theorem. After finding the distances, use the distance formula d = rt, and solve for t, t = y. The time it took for each part of the trip along a hypotenuse was the same, so the two distances divided by their respective rates are equal to one another. Because Katie paddles more quickly than she walks, let the rate at which she walks be r and the rate at which she paddles be 2.5r. 

Figure 6.25. The first principal component (PC) is the line that best fits the data points by minimizing the sum of the squared distances from the points perpendicular to the PC. By the Pythagorean theorem (ei i + A 2 = ei) lt can be seen that this is equivalent to Eqn. (2). The t -score of a subject is the distance from the average point to the projection of the subject onto the PC. The k th element of the loading vector p is the cosine of the angle between the kth variable and the PC.

To fine the distance between two points, use this variation of the Pythagorean theorem ... 

Since the edges of all six spheres are exactly the same distance from the center of the octahedral hole, we can calculate the radius of this hole most easily by focusing on the four spheres whose centers form a square (Fig. 16.37). Note from Fig. 16.37 that R is the radius of the packed spheres, r is the radius of the octahedral hole, and d is the length of the diagonal of the square. From the Pythagorean theorem... 

The previous discussion subtly shifted between molecular similarity and molecular properties. It is important to elucidate the relationship between the two. If each of the molecular properties can be treated as a separate dimension in a Euclidean property space, and dissimilarity can be equated with distance between property vectors, similarity/diversity problems can be solved using analytical geometry. A set of vectors (chemical structures) in property space can be converted to a matrix of pairwise dissimilarities simply by applying the Pythagorean theorem. This operation is like measuring the distances between all pairs of cities from their coordinates on a map. 

Euclidean distance The Euclidean distance or Euclidean metric is the ordinaiy distance between two points that one would measure with a ruler and is given by the Pythagorean formula. It can be calculated using the following formula ... 

As shown in Fig. 11.3b, D represents the distance between both planes at any instant t t > 0). According to the Pythagorean theorem, we can write the following equation ... 

Pythagorean theorem or around the block distance measures. The computation of the Euclidian distance from point a (jc,y) to the mean of class 1 measured in... 

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