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Direction Cosines and Direction Ratios of Line

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

x2x1AB, y2y1AB, z2z1AB or x1x2AB, y1y2AB, z1z2AB

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

x2x1, y2y1, z2z1

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 = (4116, 6116, 8116).

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