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 $$\vec{AB}$$ or, $$\vec{BA}$$ are the direction cosines of line L. Thus, if $$\alpha$$, $$\beta$$, $$\gamma$$ 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, $$cos\alpha$$, $$cos\beta$$, $$cos\gamma$$ or – $$cos\alpha$$, – $$cos\beta$$, – $$cos\gamma$$.

Therefore, if l, m, n are direction cosines of a line, then -l, -m, -n are also its direction cosines and we always have

$$l^2 + m^2 + n^2$$ = 1

If A$$(x_1, y_1, z_1)$$ and B$$(x_2, y_2, z_2)$$ are two points on a line L, then its direction cosines are

$$x_2 – x_1\over AB$$, $$y_2 – y_1\over AB$$, $$z_2 – z_1\over AB$$ or $$x_1 – x_2\over AB$$, $$y_1 – y_2\over AB$$, $$z_1 – z_2\over AB$$

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$$(x_1, y_1, z_1)$$ and B$$(x_2, y_2, z_2)$$ are two points on a line L, then its direction ratios are proportional to

$$x_2 – x_1$$, $$y_2 – y_1$$, $$z_2 – z_1$$

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 = $$\sqrt{16 + 36 + 64}$$ = $$\sqrt{116}$$

Direction Cosines = ($$4\over \sqrt{116}$$, $$6\over \sqrt{116}$$, $$8\over \sqrt{116}$$).