find 2 vectors in opposite directions that are orthogonal to the vector <5,6,.3>
Well, you know that the cross product between two vectors is orthogonal to both vectors.
So, pick a random vector such as k = (0,0,1) and take the cross product between v and k. Your resultant vector is orthogonal to both vectors. Furthermore, changing the sign of the resultant vector will not change the fact that it'll still be orthogonal to v and k.