In this post you will learn how to find direction cosines and direction ratios of line of the vector with examples.
Let’s begin –
Direction Cosines and Direction Ratios of Line
Direction cosines
The direction cosines of a line are defined as the direction cosines of any vector whose support is the given line.
It follows from the above definition if A and B are two points on a given line L, then the direction cosines of vectors →AB or, →BA are the direction cosines of line L. Thus, if α, β, γ are the angles which the line L makes with the positive direction of x-axis, y-axis and z-axis respectively, then its direction cosines are either, cosα, cosβ, cosγ or – cosα, – cosβ, – cosγ.
Therefore, if l, m, n are direction cosines of a line, then -l, -m, -n are also its direction cosines and we always have
l2+m2+n2 = 1
If A(x1,y1,z1) and B(x2,y2,z2) are two points on a line L, then its direction cosines are
x2–x1AB, y2–y1AB, z2–z1AB or x1–x2AB, y1–y2AB, z1–z2AB
Direction Ratios
The direction ratios of a line are proportional to the direction ratios of any vector whose support is the given line.
If A(x1,y1,z1) and B(x2,y2,z2) are two points on a line L, then its direction ratios are proportional to
x2–x1, y2–y1, z2–z1
Example : Find the direction cosines and direction ratios of the line whose end points are A(1, 2, 3) and B(5, 8, 11).
Solution : We have, A(1, 2, 3) and B(5, 8, 11)
Direction ratios = (4, 6, 8)
AB = √16+36+64 = √116
Direction Cosines = (4√116, 6√116, 8√116).