Space is a term we use to describe the presence of material objects in relation to one another or their movements. In other words, the notion of space is closely associated with material objects in relative rest and/or motion. We consider a point as an idealized model of a small material object or particle. When we combine small material objects or particles together, they form bigger objects. Similarly, when we combine points together, they give rise to various forms or shapes. A systematic study of such sets of points or shapes can be done by associating each of such points with an ordered set of real numbers. Such numbers are said to be coordinates of a given point on the space under consideration.
Coordinates in Space
The study of coordinate geometry begins by establishing a one-to-one correspondence between the points on a line and the real numbers. For this, we associate each point in a plane with an ordered pair of real numbers. Each point in the ordinary space (or three dimensional space) can be associated with an ordered triple of real numbers.
To start with, we consider three mutually perpendicular lines intersecting at a fixed point. This fixed point is called origin and the three lines are called coordinate axes. The coordinate axes are labelled as the x-axis, the y-axis and the z-axis.
The origin, the coordinate axes and their directions form a coordinate system. In practice, we denote the origin by $O$, the x-axis by $XOX’$, the y-axis by $YOY’$ and the z-axis by $ZOZ’$. The direction of the positive x-axis, y-axis and z-axis are chosen along the index finger, the middle finger and the thumb respectively. In other words, we can determine the directions of the coordinate axes by following the right hand thumb rule.
Now, let us associate a point with an ordered triple of real numbers. If we start from the origin $O$ and move a certain distance ($x$ units) along the x-axis, then we arrive at a fixed point $A$ on the x-axis. From $A$, if we move a certain distance ($y$ units) along a line parallel to the y-axis, then we arrive at a point $L$ on the xy-plane. The point $L$ is denoted by the ordered pair $(x,y)$.
From $L$, if we again move a distance ($z$ units) along a line parallel to the z-axis (or perpendicular to the xy-plane), we arrive at a unique point $P$ in the ordinary space or three dimensional space. This point $P$ is denoted by the ordered triple $(x,y,z)$. We may, however, proceed in the reversed order, we can start with a point and unique set of three numbers.
The lines $OX$ and $OY$ form a plane $XOY$. In the plane, z-coordinate of every point will be zero. So, this plane is also called xy-plane or $z=0$ plane. Similarly, $YOZ$ and $ZOX$ are the planes known as yz-plane, zx-plane or $x=0$, $y=0$ planes respectively.
In a plane, the coordinate axes divide the plane into four quadrants. In space, the coordinate planes divide the whole space into eight parts called octants. The signs of the coordinates of any point can be found by applying the right hand thumb rule. The following table gives a summarised picture of the signs:
|Coordinates $\downarrow$ | Octants $\rightarrow$||x-axis||y-axis||z-axis|
Locus and Equations
Points in space may or may not be scattered uniformly. They may or may not satisfy certain condition or conditions. Since points are associated with coordinates in coordinate geometry, conditions satisfied by given point or points can be considered as conditions placed on the coordinates of the points. These conditions can be expressed as equations or inequalities.
The set of those points and only those points that satisfy certain conditions is called a locus. The corresponding equations or inequalities are called the equations or inequalities of the locus. Moreover, each such equation or inequality defines a relation in such a way that there is a one-to-one correspondence between the ordered triple of numbers and the points of the locus. In short, a locus is a graph of a relation.