Solidos Analiticos

CHAPTER

I

CO-ORDINATES
In plane the position of a point is determined by
obtained with reference to two straight lines in the
plane generally at right angles. The position of a point in space is,
however, determined by three numbers x, y, z. We now proceed to
explain as to how this is done.
Introduction.

two numbers

x, y,

Co-ordinates of a point in space. Let X'OX, Z'OZ betwo
lines.
Through 0, their point of intersection,

1-1.

perpendicular straight

Z

Y'

e

M

X
O

Y
Fig.

1

XOZ

Y'OY perpendicular to the
three mutually perpendicular straight lines

called

the origin, draw a line

so that

we have

known

as rectangular co-ordinate axes.

plane

X'OX, TOY, Z'OZ

XOZ

(The plane
containing
the lines X'OX and Z'OZmay be imagined as the plane of the paper
the line OY as pointing towards the reader and OY' behind the paper).
The positive directions of the axes are indicated by arrow heads.
These three axes, taken in pairs, determine three planes,
;

XOY, YOZ and ZOX

ZX

or briefly XY, YZ,
planes mutually at right angles,
rectangular co-ordinate planes.

known

as

Through any point, P, inspace, draw three planes parallel to the
three co-ordinate planes (being also perpendicular to the corresponding
axes) to moot the a.xes in A B, C.
y

Let

QA=x, OB=y and

0(7=?,

ANALYTICAL SOLID GEOMETRY

2

These three numbers,

determined by the point P, are

x, y, z,

called the co-ordinates of P.

Any one of these x, y, z, will be positive or negative according
asit is measured from O, along the corresponding axis, in the positive
or negative direction.
three numbers,

Conversely, given

whose co-ordinates are
(f)

x, y, z.

To do

x, y, z, we can find a point
this, we proceed as follows :

Measure OA, OB, 00, along OX, 07,

OZ

equal to

x, y, z

respectively.
(ii)

Draw through

ZX

planes YZ,

and

A, B,

C

planesXY respectively.

parallel to

The point where these three planes

the co-ordinate

intersect is the required

point P.
Note. The three co-ordinate planes divide the whole space in eight compartments which are known as eight octants and since each of the co-ordinates
of a point may be positive or negative, there are 2 3 = 8) points whose co-ordinates have the same numerical valuesand which lie in the eight octants, one in
(

each.

Further explanation about co-ordinates.

1*11.

In

1*1

above,

we have learnt that in order to obtain the co-ordinates of a point P,
we have to draw three planes through P respectively parallel to the

three co-ordinate planes. The three planes through P and the three
co-ordinate planes determine a parallelepiped whoseconsideration
leads to three other useful constructions for determining the coordinates of P.

The

parallelopiped, in question, has six rectangular faces

PMAN, LCOB PNBL, MAOC PLCM, NBOA
;

;

(See Fig. 1).
(i) We have
x=OA = CM=LP = perpendicular

y=OB=ANMP

z=OC=AM=NP
Thus

from
perpendicular from
= perpendicular from

P on the YZ plane
P on the ZX plane
P on the XY plane.;

;

the co-ordinates x, y ; z of any point P, are the perpendicular
and
from the three rectangular co-ordinate planes YZ,

distances of

ZX

P

XY respectively.
(ii)

the line

Since

OA*

9

PA

lies in

the plane

PMAN which

is

perpendicular to

therefore

PBOB and PC

Similarly

OC.

P

Thus

the co-ordinates x, y, z of any point
arealso the distances
the origin
of the feet A, B,
of the perpendiculars from the point

C

from

to the co-ordinate

*
plane.

A

line

axes

X'X, Y'Y and Z'Z

perpendicular to a plane

is

respectively.

perpendicular to every }jne in the

DISTANCE BETWEEN

What

Ex.

co-ordinate axes

We

(Hi)

POINTS

3

are the perpendicular distances of a point (x t...