Length of a Line Segment in Three Dimensions
Date: 04/16/2004 at 13:58:26 From: Catherine Subject: Length of a line segment in three dimensions What is the length of a line segment in three dimensions with endpoints (1, 0, 2) and (1, 4, 5)?
Date: 04/16/2004 at 16:00:33 From: Doctor Douglas Subject: Re: Length of a line segment in three dimensions Hi, Catherine. Thanks for writing to the Math Forum. You can apply the Pythagorean Theorem twice in succession. Consider the box below: B-------C We want to know the distance between A and G. /| /| We know the length (AD), height (DH) and width (HG). A-------D | | F-----|-G The vertices AHG form a right triangle, because |/ |/ the segment GH is perpendicular to the plane ADHE. E-------H We apply the Pythagorean theorem first to triangle AHG, and in doing so, find we need to apply it again to get the distance AH: (AG)^2 = (GH)^2 + (AH)^2 Pythagoras (AHG) = (GH)^2 + [(AD)^2 + (DH)^2] Pythagoras (ADH) = width^2 + length^2 + height^2 The body diagonal distance (AG) is simply the square root of this, or distance = sqrt[width^2 + length^2 + height^2]. Let's think of the length of the box in terms of the x-coordinates of the points, the height in terms of y, and the width in terms of z. In other words, think of the box as being on this axis system: y z | / |/ -----+----- x /| / | Since each of those three distances is one-dimensional, we can think of the length of the box as the difference in the two x-coordinates, the width as the difference in the two z-coordinates, and the height as the difference in the two y-coordinates. Thus, modifying the above distance formula: distance = sqrt[width^2 + length^2 + height^2] distance = sqrt[(z2 - z1)^2 + (x2 - x1)^2 + (y2 - y1)^2] In other words, we've discovered that by using the Pythagorean theorem twice, we can find the distance between any two points (x1,y1,z1) and (x2,y2,z2) in three dimensional space by calculating: distance = sqrt[(z2 - z1)^2 + (x2 - x1)^2 + (y2 - y1)^2] Are you familiar with the distance formula in two dimensions? Given two points (x1,y1) and (x2,y2), the distance between them is: distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2] Can you see how the three dimensional formula is a logical extension of this formula? Now, can you use the three dimensional distance formula to find the distance between your two points? - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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