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Coordinate Planesx

In two dimensional geometry, to locate the position of a point in a plane, we consider two mutually perpendicular lines called the coordinate axes. In actual life, we not only deal with points in a plane but also points in space. To locate the position of such points consider three mutually perpendicular planes intersecting at the point O.
 
These three planes intersect along the lines X′OX, Y′OY and Z′OZ, called the x, y and z-axes, respectively. The rectangular coordinate system in three dimensional geometry is made of these three lines. The XOY plane is called the XY-plane. Similarly YOZ and ZOX planes are called YZ-plane and ZX-plane respectively. The planes are known as the three coordinate planes. Their point of intersection O is called the origin.
 
The distance measured along OX, OY and OZ are taken to be positive and those measured along OX', OY' and OZ' are taken to be negative.

In two dimensional geometry the two axes divide the plane into four quadrants. In three dimensional geometry the three planes divide space into eight parts known as octants. The octants are XOYZ, X′OYZ, X′OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′. They are denoted by I, II, III, ..., VIII respectively.
 

Coordinates of Points in Space


Now that we have a coordinate system in space, with coordinate axes, coordinate planes and origin, we are going to see how to associate three numbers(coordinates) for a given point. Also given a triplet, let us see how to locate the point in the coordinate system.

From the fixed point we drop a perpendicular PM on the XY plane with M as the foot of the perpendicular. Now, from M draw a perpendicular ML to the X axis, meeting it at L. Let OL be , LM be and MP be . Then are called the coordinates, respectively, of the point P in the space.

In the above diagram, notice that the point P lies in the first octant (XOYZ) and so its coordinates are positive. But if the point lies in any other octant then the signs of the coordinates will change accordingly.Thus, to each point P in the space there corresponds an ordered triplet of real numbers.

Conversely if a triplet is given, we locate the point using the method shown in the following visual.

 

Thus the point so obtained has the coordinates . Note that LM is perpendicular to the X axis or is parallel to the Y axis. PM is perpendicular to the XY plane. Thus there is a one to one correspondence between the ordered triplet of real numbers and the points in space.if P is any point in the space, then  are perpendicular distances from YZ, ZX and XY planes, respectively as shown in the visual below.

The coordinates of the origin O are (0, 0, 0).

The coordinates of any point on the will be as .

The coordinates of any point on the will be as .

The coordinates of any point on the will be as .

The coordinates of any point in the XY-plane will be as .

The coordinates of any point in the YZ-plane will be as .

The coordinates of any point in the ZX-plane will be as .

The sign of the coordinates of the point depends on the octant in which the point lies.

The following table shows the signs of the coordinates in eight octants.


Example 1:
In the following diagram if D is the point (3, 4, 5), then find the coordinates of A, B, C, E, F and G.

Solution:
If D is the point (3, 4, 5) we have OA = 3, OC = 4 and OF = 5.
Since OABCEFGD is a parallelopiped we have
OA = BC = DE = GF == 3
OC = AB = DG = EF = = 4
OF = AG = BD = CE = = 5
Now A is a point on the X axis .
The coordinates of A is of the form i.e., (3, 0, 0).

B is a point on the XY plane.
The coordinates of B is of the form i.e., (3, 4, 0).

C is a point on the Y axis.
The coordinates of C is of the formi.e., (0, 4, 0).

E is a point on the YZ plane.
The coordinates of E is of the form i.e., (0, 4, 5).

F is a point on the Z axis.
The coordinates of F is of the form i.e., (0, 0, 5).

G is a point on the XZ plane.
The coordinates of G is of the form i.e., (3, 0, 5).

Example 2:
Name the octants in which the following points lie. ( 1, 5, 7), ( 2, 4, 3), ( 4, 1, 3), (1, 2, 3), (5, 3, 2), (8, 6, 2), (3, 2, 7), ( 4, 2, 3).

Solution:
Consider the point ( 1, 5, 7). Since all the coordinates are negative, it lies in octant VII.
Consider the point ( 2, 4, 3). Since the and coordinates are negative, it lies in octant III.
Consider the point ( 4, 1, 3). Since the and coordinates are negative, it lies in octant VI.
Consider the point (1, 2, 3). Since all the coordinates are positive, it lies in octant I.
Consider the point (5, 3, 2). Since the and coordinates are negative, it lies in octant VIII.
Consider the point (8, 6, 2). Since the coordinate alone is negative, it lies in octant IV.
Consider the point (3, 2, 7). Since the coordinate alone is negative, it lies in octant V.
Consider the point ( 4, 2, 3). Since the coordinate alone is negative, it lies in octant II.

Example 3:
Name the axis or plane in which the following points lie. (0, 3, 0), (0, 4, 5), (2, 0, 7), (4, 0, 0), (0, 0, 6) and (1, 7, 0).

Solution:
Consider the point (0, 3, 0). Since the and coordinates are zero, it lies on the Y axis.
Consider the point (0, 4, 5). Since the coordinate alone is zero, it lies in the YZ plane.
Consider the point (2, 0, 7). Since the coordinate alone is zero, it lies in the XZ plane.
Consider the point (4, 0, 0). Since the and coordinates are zero, it lies on the X axis.
Consider the point (0, 0, 6). Since the and coordinates are zero, it lies on the Z axis.
Consider the point (1, 7, 0). Since the coordinate alone is zero, it lies in the XY plane.




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