We can get a unit vector, by adding three unit vectors. The only condition needs to be satisfied is that two of the vectors should be in opposite direction so that they cancel out each other and the resultant vector is third unit vector

If any two unit vectors are along the co-ordinate axes, then by taking the third unit vector opposite to any of the other two vectors and along the co-ordinate axes it will cancel out each other thus we will have a resultant unit vector left over.