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Distance of Point from Origin
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You might ask, Can the distance of a point from the origin in Cartesian three-space ever be undefined Can it ever be negative The answers are no, and no! Imagine a point P in Cartesian three-space anywhere you want with the coordinates (xp,yp,zp) To find the distance of P from the origin, you start by squaring xp, which is the x coordinate of P Because xp is a real number, its square is a nonnegative real Then you square yp, which is the y coordinate of P This result must also be a nonnegative real Then you square zp, which is the z coordinate of P This square, too, is a nonnegative real Next, you add the three nonnegative reals xp2, yp2, and zp2 That sum must be another nonnegative real Finally, you take the nonnegative square root of the sum of the squares The nonnegative square root of a nonnegative real number is always defined; and it s never negative itself, of course!
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The formula we derived here is based on the idea that we start at the origin and go outward to point P If we go inward from P to the origin, the distance is exactly the same (If we were working with vectors, the vector displacements would be negatives of each other, but we re not there yet)
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Suppose we re given a point P = (xp,yp,zp) in Cartesian three-space Prove that if we negate any one, any two, or all three of the coordinates, the resulting point is the same distance from the origin as P
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For the point P, the distance c from the origin is c = (xp2 + yp2 + zp2)1/2 The square of any real number is always the same as the square of its negative That tells us three things: ( xp)2 = xp2 ( yp)2 = yp2 ( zp)2 = zp2 By substitution, all these quantities are identical: (xp2 + yp2 + zp2)1/2 [( xp)2 + yp2 + zp2]1/2 [xp2 + ( yp)2 + zp2]1/2 [xp2 + yp2 + ( zp)2]1/2 [( xp)2 + ( yp)2 + zp2]1/2 [( xp)2 + yp2 + ( zp)2]1/2 [xp2 + ( yp)2 + ( zp)2]1/2 [( xp)2 + ( yp)2 + ( zp)2]1/2
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Cartesian Three-Space
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These quantities represent the distances of the following points from the origin, respectively: (xp,yp,zp) ( xp,yp,zp) (xp, yp,zp) (xp,yp, zp) ( xp, yp,zp) ( xp,yp, zp) (xp, yp, zp) ( xp, yp, zp) That s all the points we can get, in addition to P itself, by negating any one, any two, or all three of the coordinates of P They re all the same distance c from the origin, where c = (xp2 + yp2 + zp2)1/2
Distance between Any Two Points
When we want to determine the distance between any two points in Cartesian three-space, we can expand the formula from Cartesian two-space that we learned in Chap 1 into an extra dimension
The general formula Imagine two different points in Cartesian three-space, after the fashion of Fig 7-5 Let s call the points and their coordinates
P = (xp,yp,zp) and Q = (xq,yq,zq) where each coordinate can range over the entire set of real numbers The distance d between these points, as we follow a straight-line path from P to Q, is d = [(xq xp)2 + (yq yp)2 + (zq zp)2]1/2 If we start at Q and finish at P, we reverse the orders of subtraction, so the formula becomes d = [(xp xq)2 + (yp yq)2 + (zp zq)2]1/2 We always subtract starting coordinates from finishing coordinates
Distance between Any Two Points
+z Q (xq, yq, zq)
What s the straight-line distance d between points P and Q
d x +x
P (xp, yp, zp)