Three Dimensional geometry

Intorduction

In earlier classes we studied about the coordinates in two planes that is the XY plane. Here we are going to study in detail about the coordinates in three planes that is X, Y and the Z planes. First let us study about the direction cosines and direction ratios of lines.

1.     If the line makes angles α β γ with the positive directions of x- axis, y-axis and z-axis respectively then cos α , cos β , cosγ arecalled its direction cosines and are usually denoted as l,m,n.

2.     Direction cosines of x axis are 1,0,0

3.     Direction cosines of y-axis are 0,1,0

4.     Direction cosines of z-axis are 0,0,1.

Direction Ratios of a line.

1.     3 numbers a,b,c are called direction ratios of a line if l/a = m/b = n/c, where l,m,n are the direction cosines of the line.

2.     If l,m,n are the direction ratios of a line then l² + m² + n² = 1.

3.     If a line makes  α β γ  with the positive direction of x,y,z axes respectively then cos²α + cos²β + cos² γ = 1 and

4.     Sin²α+sin²β + sin²  γ  = 2.

6.     Direction ratios of the line joining A(x₁,y₁,z₁) and B(x₂,y₂,z₂) are x₁-x₂,

y₁-y₂, z₁-z₂ or vice versa.

Direction cosines of the line joining A(x₁,y₁,z₁) and B(x₂,y₂,z₂) are

x₁-x₂/AB, y₁-y₂/AB _, z₁-z₂/AB