# Coordinate System in Three Dimensions  ## Coordinate System in Three Dimensions:

We live in a world of three dimensions and so we should extend our knowledge of geometry to three-dimensional space. To specify the position of a point in space, we require three axes of reference- X-axis, Y-axis, and Z-axis.

In the figure, X’OX, Y’OY, and Z’OZ are the three axes of references. The positive directions of the axes have been indicated by arrowheads. The axes are perpendicular to each other and so they are referred to as the rectangular axes. The point of intersection of the axes is the origin. In the figure, O is the origin.

### Coordinates Planes:

Each pair of axes constitutes a plane. The X-axis and Y-axis are on a plane called the XY-plane. Similarly, the plane containing Y-axis and Z-axis is called the YZ-plane and the plane containing Z and X-axes is called the ZX-plane. The plane XOY, YOZ, and ZOX are known as coordinate planes. These planes divide the entire space into eight parts known as the octans.

### Coordinates of a Point:

In order to specify the position of a point on a plane we require an ordered pair of real numbers. In three-dimensional space, we require a triplet of real numbers to describe a point. Each number denotes the distance of the point from a coordinate plane. Thus, if P be a point in space, its coordinates may be denoted by (x, y, z), where x denotes the distance of the point from the YZ-plane, y is the distance from the ZX-plane and z from the XY-plane. The ordered triplet of real numbers constitutes cartesian coordinates of the point P and is written as P (x, y, z).

Thus, in the figure, OA = x, OB = y, and OC = z. To have a clear idea about the geometry in space, the rectangular parallelepiped has been completed with OA, OB, and OC as coterminous edges through the origin O and P at the opposite vertex.

### Distance Formula:

Let P (x1, y1, z1) and Q (x2, y2, z2) be two given points in space. Through P and Q, planes are drawn parallel to the coordinates planes to form a cuboid with the line segment PQ as one of its diagonals as shown in the figure. QL is perpendicular to the plane PCLA and PL lies in this plane. So, QL ⊥ PL. ∴ ∠PLQ = 90°.

### Section Formula:

Coordinates of a point that divides the line segment joining two points in a given ratio. Let P (x1, y1, z1) and Q (x2, y2, z2) be the two given points. Let R (x, y, z) be the point that divides PQ internally in the ratio m : n, so that PR : RQ = m : n. PL, QM, and RN are drawn perpendicular to the ZX plane. Through R, SRT is drawn parallel to LNM to meet LP produced at S and QM at T as shown in the figure.

### Note:

(i) If m : n is negative, then the segments PR and RQ have opposite directions and in that case, R divides PQ externally in the ratio m : n as shown in the figure.

(ii) To find out the ratio in which a given point divides the join of two given points, it is convenient to consider the ratio λ : 1. (instead of inviting two unknowns m and n). The coordinates of R then become [(λx2 + x1)/(λ + 1), (λy2 + y1)/(λ + 1), (λz2 + z1)/(λ + 1)].

(iii) If R divides PQ internally in the ratio 1 : 1, then R is the mid-point of PQ. In that case,

Thus, the coordinates of the mid-point of the line segment joining the points (x1, y1, z1) and (x2, y2, z2) are [(x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2].

### Coordinates of the Centroid of a Triangle:

Let A (x1, y1, z1), B (x2, y2, z2), and C (x3, y3, z3) be the vertices of a given triangle as shown in the figure.

If D is the mid-point of the side BC, then the coordinates of D are [(x2 + x3)/2, (y2 + y3)/2, (z2 + z3)/2].

The centroid G divides the median AD internally in the ratio 2 : 1. If (x,