Knowing spatial coordinates of two points in any system it is possible to determine without difficulties straight line piece length between them. It is described below as to make it in relation to the two-dimensional and three-dimensional Cartesian (rectangular) system of coordinates.
Instruction
1. If coordinates of extreme points of a piece are given in the two-dimensional system of coordinates, then having drawn the straight lines perpendicular to axes of coordinates through these points, you receive a rectangular triangle. The initial piece will be its hypotenuse, and legs form pieces which length is equal to a hypotenuse projection to each of coordinate axes. From Pythagorean theorem defining a hypotenuse length square as the sum of squares of lengths of legs it is possible to draw a conclusion that for finding of length of an initial piece it is enough to find lengths of two of its projections to coordinate axes.
2. Find lengths (X and Y) projections of an initial piece to each axis of a system of coordinates. In a two-dimensional system each of extreme points is presented by couple of numerical values (X1; Y1 and X2; Y2). Lengths of projections are calculated by finding of a difference of coordinates of these points on each axis: X = X2-X1, Y = Y2-Y1. It is possible that one or both received values will be negative, but in this case it does not play any role.
3. Calculate length of an initial piece (A), having found a square root from the sum of squares of lengths of projections to axes of coordinates calculated on the previous step: A = √ (X²+Y²) = √ ((X2-X1)²+(Y2-Y1)²). For example, if the piece is carried out between points with coordinates 2;4 and 4;1, then its length will be equal √ ((4-2)²+ (1-4)²) = √13 ≈ 3.61.
4. If coordinates of the points limiting a piece are given in the three-dimensional system of coordinates (X1; Y1; Z1 and X2; Y2; Z2), a formula of finding of length (A) of this piece will be similar received on the previous step. In this case it is necessary to find a square root from the sum of squares of projections to three coordinate axes: A = √ ((X2-X1)²+(Y2-Y1)²+(Z2-Z1)²). For example, if the piece is carried out between points, with coordinates 2;4;1 and 4;1;3, then its length will be equal √ ((4-2)²+ (1-4)²+ (3-1)²) = √17 ≈ 4.12.