
In three dimensions, the coordinate axes of a rectangular Cartesian coordinate system are three mutually perpendicular lines. The axes are called the x, y and zaxes.

The three planes determined by the pair of axes are the coordinate planes, called XY, YZ and ZXplanes.

The three coordinate planes divide the space into eight parts known as octants.

The coordinates of a point P in three dimensional geometry is always written in the form of triplet like (x, y, z). Here x, y and z are the distances from the YZ, ZX and XYplanes.

Any point on xaxis is of the form (x, 0, 0)

Any point on yaxis is of the form (0, y, 0)

Any point on zaxis is of the form (0, 0, z).

Distance between two points P(x1, y1, z1) and Q (x2, y2, z2) is given by
PQ =

The coordinates of the point R which divides the line segment joining two points
and Q (
) internally and externally in the ratio m : n are given by
and

The coordinates of the midpoint of the line segment joining two points P(
) and Q(
) are
[(x1+x2)/2, (y1+y2)/2, (z1+z2)/2]

The coordinates of the centroid of the triangle, whose vertices are (
) (
) and (
), are
[, , ]
Examples
Question
Find the distance between the points P(1, –3, 4) and Q (– 4, 1, 2).
Solution
The distance PQ between the points P (1,–3, 4) and Q (– 4, 1, 2) is
PQ = (41)^{2 }+ (1(3))^{2 }+ (24)^{2}
= 45 units
Question
Find the equation of the set of the points P such that its distances from the points A (3, 4, –5) and B (– 2, 1, 4) are equal.
Solution
If P (x, y, z) be any point such that PA = PB
(x − 3)^{2} + (y − 4)^{2} + (z + 5)^{2} = (x + 2)^{2} + (y −1)^{2} + (z − 4)^{2}
(x − 3)^{2} + ( y − 4)^{2} + (z + 5)^{2} = (x + 2)^{2} + (y −1)^{2} + (z − 4)^{2}
10 x + 6y – 18z – 29 = 0
Question
The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, –5, 7) and (–1, 7, – 6), respectively, find the coordinates of the point C.
Solution
Let the coordinates of C be (x, y, z) and the coordinates of the centroid G be (1, 1, 1). Then
= 1
x = 1
= 1
y = 1
= 1
z = 2