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3D GEOMETRY – PROPERTIES EXPLAINED

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.


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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 (x1, y1, z1) and B (x2, y2, z2 ) in the ratio m:n, then

x =  ,   y =  , z = .

  • The midpoint of the line joining A (x1, y1, z1) and B (x2, y2, z2 ) is given by

M {(x1 + x2)/2 , (y1 + y2)/2 , (z1 + z2)/2 }

SOME RESULTS ON LINES IN SPACE

  1. 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 .

  1. 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.

  1. ANGLE BETWEEN TWO LINES

If ? is the angle between two lines, L1 and L2 whose direction cosines are l1, m1 ,n1 and l2, m2 ,n2 respectively, hen the following equations hold true.

  • Cos ? = l1.l2 + m1.m2 + n1.n2
  • Sin ? =
  • Lines L1 and L2 are perpendicular ? l1.l2 + m1.m2 + n1.n2 = 0.
  • Lines L1 and L2 are parallel ?   =  =  .