This article consists of some fundamental concepts of three dimensional geometry.

COORDINATES OF A POINT IN SPACE1

Let O be the origin, and let OX, OY and OZ be three mutually perpendicular lines taken as the x-axis, y-axis and z-axis respectively in such a way that they form a right handed system.

The planes YOZ, ZOX, XOY are respectively knows as the yz-plane, the zx plane and the xy plane.

These planes, called coordinate planes divide the space into eight pots called octants.

Position vector of a point in space.

Let , , be unit vectors OX, OY and OZ respectively.

If P(x, y, z) is a point in space, we say that the position vector of P is ( x + y + z ).

SOME RESULTS ON POINTS IN SPACE

- DISTANCE BETWEEN TWO POINTS

The distance between two points P (a_{, }b_{,} c) and Q ( x, y, z ) is given by

PQ =

- SECTION FORMULA
- (i) If P(x, y, z) divides the join of A (x
_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2}) in the ratio m:n, then

x = , y = , z = .

- The midpoint of the line joining A (x
_{1}, y_{1}, z_{1}) and B (x_{2}, y_{2}, z_{2}) is given by

M {(x_{1} + x_{2})/2 , (y_{1 }+ y_{2})/2 , (z_{1} + z_{2})/2 }

SOME RESULTS ON LINES IN SPACE

- DIRECTION COSINES OF A LINE:

If a line makes angles ?, ?, ? with the x-axis, y-axis and z axis respectively then

L=cos ?, m = cos ? and n= cos ?

are called the direction cosines (or, d.c’s) of the line.

We always have .

- DIRECTION RATIOS OF A LINE

Any three numbers a, b, c proportional to the direction cosines l, m, n respectively of a line are called the direction ratios of the line.

Clearly we have l/a = m/b = n/c .

If a, b, c are the direction ratios of a line, then its direction cosines are

L = , m = , n =

If r = ( x + y + z ), then the direction ratios of r are x, y, z.

- ANGLE BETWEEN TWO LINES

If ? is the angle between two lines, L_{1} and L_{2} whose direction cosines are l_{1}, m_{1} ,n_{1} and l_{2}, m_{2} ,n_{2} respectively, hen the following equations hold true.

- Cos ? = l
_{1}.l_{2}+ m_{1}.m_{2}+ n_{1}.n_{2} - Sin ? =
- Lines L
_{1}and L_{2}are perpendicular ? l_{1}.l_{2}+ m_{1}.m_{2}+ n_{1}.n_{2 }= 0. - Lines L
_{1}and L_{2 }are parallel ? = = .