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

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