We know that how to find the angle between two lines in a plane.
* If two lines in the space are intersecting then there exists a unique plane passing through those two lines.
* In this case the angle between the lines is similar to that of in the two dimensional geometry.
* Now we define the angle between two non-intersecting lines.
* The angle between two non-intersecting lines is defined as the angle between the two lines drawn parallel to them through any point in the space.
* If a given directed line (ray) makes angles α, β, γ with positive direction of the axes of x, y and z respectively then cos α, cos β, cos γ are called the direction cosines (d.c s) of the line and they are denoted by l, m, n.
i.e. l = cos α m = cos β n = cos γ
* If l, m, n are direction cosines of a line then -l, -m, -n are also direction cosines of the line. Usually we take one trial of direction cosines l, m, n and these are denoted by the ordered triple (l, m, n).
Since the line makes with the positive directions of x - axis, y- axis, z-axis, the angles 0°, 90°, 90° respectively.
* Then the direction cosines of x-axis are (cos0°, cos90°, cos 90°) i.e. (1, 0, 0) Similarly (0, 1, 0) and (0, 0, 1) are the direction cosines of y and z - axes respectively.
Let α,β,γ be the angels made by the directed line with the positive direction of x-axis, y-axis, z - axis respectively such that l = cos α, m = cos β, n = cos γ
Let (x, y, z) be the coordinates of P
Let M be the projection of P on x - axis.
Then OM = x
x = l r
Similarly y = mr
and z = nr
P = (lr, mr, nr)
Let P be a point on the line Parallel to the given line and passing through the origin such that OP = 1
Then, P = (l,m,n)
Now OP = 1
2+m2+n2 = 1
cos2 α + cos2 β + cos2 γ = 1 Where α, β, γ are the angles made by adirected line with the positive direction of coordinate axes.
Three real numbers a,b,c are said to be direction ratios (or) direction numbers of a line if a:b:c = l:m:n where (l,m,n) are d.c.'s of the line.
a : b : c = l : m : n
