When direction cosines of a line are calculated as (per their definition states) the 'cosines' of the angles made by the line with positive X, Y, Z axis (angles measured counterclockwise), we get a unique set of cosines, yes. (unique for all the parallel lines actually)
But often in co-ordinate geometry, we only have the information of two particular points on the line, and depending on the direction/order of points that we consider, we get two sets of direction cosines, one just opposite in sign of the other.
This wouldn't be a problem except that there does exist a totally another line with the opposite-in-sign direction cosines of our first line which is not parallel to it but only makes the complement of the angles with the coordinate axes (i.e. theta and 180-theta).
This problem doesn't appear with vectors because they are directional of course.
But with lines, if I am given a set of direction ratios or two particular points on it, how will I know which one of the two possible sets of direction cosines is the true one?
(Sorry it's all text, I don't know latex)