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I perfectly understand the concept of direction cosines for a Vector.

Corresponding to an axis, the direction cosine is just the cosine of the angle that the Vector makes with the positive direction of the axis.

But I have problems in uniquely defining the direction cosines of a Line. According to me there is no way to uniquely specify direction cosines of a Line since it has no unique direction. There should always be 2 sets of direction cosines (of opposite sign).

There was a question in my textbook that read: " If a line is parallel to $-18i+12j-4k$ then what are its direction cosines?"

In the solution they have simply calculated the direction cosines of the Vector given, Why isn't the negative counterpart of the direction cosines also given as the solution?

Another example, the direction cosines of the $x$ axis is always given as $(1,0,0)$. But $(-1,0,0)$ is also valid according to me.

Is there some convention we have to follow?

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The convention followed is usually this: Direction cosines are cosines of angles with positive axes measured counterclockwise. Let $\alpha_x, \alpha_y, \alpha_z$ be angles made by a line with positive x, y, z axes respectively in counterclockwise sense, then DCs are $\cos \alpha_x, \cos \alpha_y, \cos \alpha_z$.

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    $\begingroup$ Thank You for your answer! $\endgroup$ Mar 21, 2021 at 4:13

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