8.2 Three{dimensional space

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DEFINITION 8.2.1 Three{dimensional space is the set E3 of ordered

triples (x; y; z), where x; y; z are real numbers. The triple (x; y; z) is called

a point P in E3 and we write P = (x; y; z). The numbers x; y; z are called,

respectively, the x; y; z coordinates of P.

The coordinate axes are the sets of points:

f(x; 0; 0)g (x{axis); f(0; y; 0)g (y{axis); f(0; 0; z)g (z{axis).

The only point common to all three axes is the origin O = (0; 0; 0).

The coordinate planes are the sets of points:

f(x; y; 0)g (xy{plane); f(0; y; z)g (yz{plane); f(x; 0; z)g (xz{plane).

The positive octant consists of the points (x; y; z), where x > 0; y >

0; z > 0.

We think of the points (x; y; z) with z > 0 as lying above the xy{plane,

and those with z < 0 as lying beneath the xy{plane. A point P = (x; y; z)

will be represented as in Figure 8.3. The point illustrated lies in the positive

octant.

DEFINITION 8.2.2 The distance P1P2 between points P1 = (x1; y1; z1)

and P2 = (x2; y2; z2) is de¯ned by the formula

P1P2 =

p

(x2 ¡ x1)2 + (y2 ¡ y1)2 + (z2 ¡ z1)2:

For example, if P = (x; y; z),

OP =

p

x2 + y2 + z2:

8.2. THREE{DIMENSIONAL SPACE 155

¢

¢

¢

¢

¢

¢®

-

6

y

z

x

OQ

Q

Q

Q

Q

(x; 0; 0) Q¢(x; y; 0)

¢

¢

¢

(0; y; 0)

Q

Q

Q

Q

Q

Q

(0; 0; z)

P = (x; y; z)

Figure 8.3: Representation of three-dimensional space.

- y

·

·

·

·

·

·

·

·

·

·À x

(x2; 0; 0)

(x1; 0; 0)

(0; y1; 0) (0; y2; 0)

(0; 0; z1)

(0; 0; z2)

6

z

b

b

b

b

b

b

b

b

b

b

¢

¢

¢

¢

¢

¢

b

b

b

b

b

b

b

b

b

b

T

T

T

T

T

·

·

·

·

·

T

T

T

T

T

A

B

Figure 8.4: The vector

-

AB.

156 CHAPTER 8. THREE{DIMENSIONAL GEOMETRY

DEFINITION 8.2.3 If A = (x1; y1; z1) and B = (x2; y2; z2) we de¯ne

the symbol

-

AB to be the column vector

-

AB=

2

4

x2 ¡ x1

y2 ¡ y1

z2 ¡ z1

3

5 :

We let P =

-

OP and call P the position vector of P.

The components of

-

AB are the coordinates of B when the axes are

translated to A as origin of coordinates.

We think of

-

AB as being represented by the directed line segment from

A to B and think of it as an arrow whose tail is at A and whose head is at

B. (See Figure 8.4.)

Some mathematicians think of

-

AB as representing the translation of

space which takes A into B.

The following simple properties of

-

AB are easily veri¯ed and correspond

to how we intuitively think of directed line segments:

(i)

-

AB= 0 , A = B;

(ii)

-

BA= ¡

-

AB;

(iii)

-

AB +

-

BC=

-

AC (the triangle law);

(iv)

-

BC=

-

AC ¡

-

AB= C ¡ B;

(v) if X is a vector and A a point, there is exactly one point B such that

-

AB= X, namely that de¯ned by B = A + X.

To derive properties of the distance function and the vector function -

P1P2, we need to introduce the dot product of two vectors in R3.

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