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My question is can the angles be taken arbitrarily or they should satisfy some condition like angles in a triangel add up to pi radians.

I have a dilemma. The method we see in books indicates like we can arbitrarily choose the angles.

But let us consider the following:

The vectors making gamma angle with z axix should necessarily be on a cone. Now the condition that it should make alpha with x axis restrains this vector space to vectors on the cone to a few that make alpha angle angel with x-aix. Out of these vectors are we always sure that there will be a vector which also makes beta angle with the y-axix, whatever be the alpha, beta and gamma?