Coordinates of a Point in Three Dimensions

In order to locate the position of point in a space, we require a rectangular coordinate system. After choosing a fixed co-ordinate system in 3D, the coordinates of any point P in that system can be given by an ordered 3-tuple (x , y , z). Also, if the co-ordinates ( x , y ,z) are already known then we can easily locate the point P in space.

Let there be a point P in space as shown in the figure below. If we drop a perpendicular PB on the XY plane and then from the point B, we drop perpendiculars BA and BC on x – axis and y – axis respectively. Assuming the length of the perpendiculars BC, BA and PB as x, y and z respectively. These lengths x, y and z are known as the co-ordinates of the point P in three dimensional space. It must be noted that while giving the co-ordinates of a point, we always write them in order such that the co-ordinate of x – axis comes first, followed by the co-ordinate of y-axis and then z – axis. Thus for each point in space there exists an ordered 3-tupleof real numbers for its representation.

In the figure given above the co-ordinates of P are given by (x , y , z).The co-ordinates of the origin O is (0 , 0 , 0) Also the co-ordinates of the point A is given by (x , 0 , 0)as A lies completely on the x – axis. Similarly, the co-ordinates of any point on y – axis is given as (0 , y , 0) and on z – axis the coordinates are given as (0 , 0 , z). Also the co-ordinates of any point in three planes XY , YZ and ZX will be (x,y,0) , (0,y,z) and (x,0,z) respectively.

In questions, where we are asked to locate a point,i.e. when the co-ordinates of the point are given, then we have to draw three planes parallel to XY , YZ and ZX plane meeting the three axes in points A,B and C as shown in the figure. Let OA = x , OB = y and OC = z. Then the co-ordinates of the point are given as (x,y,z).

The planes ADPF, BDPE and CEPF intersect at point P which corresponds to the ordered triplet ( x , y , z).

To determine the octant in which a point lies, the signs of the co-ordinates of a point are helpful. The following table depicts the sign of the co-ordinates of a point and the octant in which it lies.

x + + + +
y + + + +
z + + + +

Using the above table we can easily figure out the signs of co-ordinates of a point or the octant in which it lies.