Perhaps you have written the problem wrongly. Are you, perhaps, given that u and v are independent vectors in S1 and to find a third vector that extends the set to a basis for ? In that case, any vector that is NOT in S1 will do but the simplest to get would be the vector perpendicular to S1, i- j- 3k.
(Well, perhaps not the simplest. None of i= 1i+0j+ 0k so x= 1, y= 0, z=0, j= 0i+ 1j+ 0k so x= 0, y= 1, z= 0, or k= 0i+ 0j+0k so x= 0, y= 0, z= 1, satisfy x- y= 3z so any of those will do.)