In three dimensions, the coordinate axes of a rectangular Cartesian co-ordinate system are three mutually perpendicular lines. The axes are called the x, y and z axes. The three planes determined by the pair of axes are the coordinate planes are called XY, YZ and ZX planes. The three coordinate planes divide the space into eight parts known as octants.

Co-ordinates of a Point

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 XY planes.

  • Any point on x-axis is of the form (x, 0, 0)
  • Any point on y-axis is of the form (0, y, 0)
  • Any point on z-axis is of the form (0, 0, z)

Distance Between Two Points

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

Direction Cosines

Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes. If l, m, n are the direction cosines of a line, then

l2 + m2 + n2 = 1

Direction cosines of a line joining two points P(x1, y1, z1) and Q(x2, y2, z2) are

Direction Ratios

Direction ratios of a line are the numbers which are proportional to the direction cosines of a line. If l, m, n are the direction cosines and a, b, c are the direction ratios of a line, then

Skew Lines

Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.

Angle Between Skew Lines

Angle between skew lines is the angle between two intersecting lines drawn from any point (or origin) parallel to each of the skew lines.