Many of us have seen how to find a vector satisfying the following conditions (i) magnitude is m (ii) makes angles alpha, beta and gamma with i,j,k vectors i.e. unit vectors along x,y and z axes. 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?