Signs of angle measured in 3D In a two dimensional case when angle between one of the axes and a line is to be measured, a counter clockwise rotation from the axis gives us a positive angle whereas a clockwise rotation gives us a negative angle. 
A detailed explation of the 2d case can be seen here:
http://mathonweb.com/help_ebook/html/angles.htm
My question is regarding the sign of these angles when we go to 3D.
The magnitude of the angle made with the axes can be calculated for lines by using the direction ratios concept. But what will be the sign of these angles measured from the axes? What direction of rotation will provide a positive angle? (especially the direction of rotation for angle made with z axis) 
 A: Measuring angle in 3D makes you lose a sensible method of labeling angles simply by sign: since the rotation can be in more directions than simply "clockwise" or "counterclockwise", you need more than "positive" and "negative" to cover all the bases.  What generally happens instead is your angle is considered to be around an axis.  So, for instance, there is a rotation that takes $(1,0,0)$ to $\left(0,\sqrt{\frac{1}{2}},\sqrt{\frac{1}{2}}\right)$: it's a quarter circle around $\left(0,-\sqrt{\frac{1}{2}},\sqrt{\frac{1}{2}}\right)$.
You will note that the axis here is in fact the cross product of the two initial vectors; typically you'll also want the axis to be normalized to length 1.
Some further study:
Quaternions are a mathematical construct often used to describe rotations in 3 dimensions.
Euler Angles are a method of describing arbitrary 3D rotations as three rotations around fixed axes.
A: For your question to have an answer, there needs to be an orientation chosen in the 3-D space.  (Just like the "counterclockwise" rule is just an orientation chosen, arbitrarily, in the 2-D space.)
https://en.wikipedia.org/wiki/Orientation_(geometry)
