√a2+ b2+ c2

√(x2– x1)2

√(x2– x1)2+ (y2– y1)2+ (z2– z1)2

√a2+ b2+ c2

√a2+ b2

√a2+ b2

√(a– 0)2+ (b– 0)2

√(x– a)2+ (y– b)2

142 distribution

rand center C. The distance formula gives the equation

of such a circle as r= , or (x– a)2+

(y– b)2= r2.

In three-dimensional space, the distance dbetween

two points P1= (x1,y1,z1) and P2=(x2,y2,z2) is found via

two applications of Pythagoras’s theorem. For instance,

the distance between the points O= (0,0,0) and A=

(a,b,c) is found by first noting that P= (a,b,0) is the

point directly below Alying in the xy-plane, and that

the triangle OPA is a right triangle. By the two-dimen-

sional distance formula, the distance between Oand P

is = . The length of the ver-

tical line connecting Pto Ais c, and the length of the

hypotenuse OA is the distance dwe seek. By Pythagoras’s

theorem we have: d2=

()

2+ c2= a2+ b2+ c2.

Thus: d= .

A slight modification of this argument shows that

the general three-dimensional distance formula for the

distance between two points P1= (x1,y1,z1) and P2=

(x2,y2,z2) is:

d=

The set of all points (x,y,z) in space a fixed distance r

from a given point C= (a,b,c) form a

SPHERE

with radius

rand center C. The distance formula gives the equation

of such a sphere as (x– a)2+ (y– b)2+ (z– c)2= r2.

The distance formula generalizes to points in n-

dimensional space as the square root of the sum of the

differences of the ncoordinates squared. This works even

for one-dimensional space: the distance between two

points x1and x2on the number line is d= .

This is precisely the

ABSOLUTE VALUE

|x2– x1|.

The

LENGTH

of a

VECTOR

v= < a,b,c > is given by

the distance formula: If we place the vector at location

O= (0,0,0) so that its tip lies at A= (a,b,c),then its

length is |v| = .

Distance of a Point from a Plane in

Three-Dimensional Space

The distance of a point Pfrom a plane is defined to be the

distance between Pand the point Nin the plane closest to

P. Suppose that the point Phas coordinates P= (x0,y0,z0)

and the

VECTOR EQUATION OF A PLANE

is ax + by + cz + d

= 0 where n= < a,b,c > is the normal to the plane. Then

Nis the point (x1,y1,z1) in the plane with vector

90°to the plane. This means that the vector is

parallel to n, and so = knfor some constant k. This

gives the equation <x0– x1,y0– y1,z0– z1> = k<a,b,c>,

and so x1= x0– ka,y1= y0– kb, and z1= z0– kc. Since

N= (x1,y1,z1) lies in the plane, this point also satisfies

the equation of the plane. Algebraic manipulation then

shows that . By the distance

formula, the distance between Pand Nis:

This establishes:

The distance of a point P= (x0,y0,z0) from

the plane ax + by + cz + d= 0 is given by the

formula:

Distance of a Point from a Line in

Two-Dimensional Space

The distance of a point Pfrom a line is defined to be

the distance between Pand the point Nin the line clos-

est to P. The

EQUATION OF A LINE

is a formula of the

form ax + by + c= 0. An argument analogous to the

one presented above establishes:

The distance of a point P= (x0,y0) from the

line ax + by + c= 0 is given by the formula:

See also

COMPLEX NUMBERS

.

distribution Any table or diagram illustrating the

frequency (number) of measurements or counts from

an experiment or study that fall within certain preset

categories is called a distribution. (See

STATISTICS

:

DESCRIPTIVE

.) For example, the heights of 1,000 8-year-

old children participating in a medical study can be