<< Calculus`VectorAnalysis`

v1 = {1, 1, 0};
v2  =  {0, 1, 0};
v3  =  CrossProduct[v1, v2];
v2  =  CrossProduct[v3, v1];
f1  =  {v1, v2, v3};
v1 = {1, 0, 0};
v2  =  {0, 1, 0};
v3  =  CrossProduct[v1, v2];
v2  =  CrossProduct[v3, v1];
f2  =   { v1, v2, v3};

n1  =  
        CrossProduct[f1[[1]], f2[[1]]];
n2  =  
        CrossProduct[f1[[2]], f2[[2]]];
n3  =  CrossProduct[f1[[3]], f2[[3]]];

VectorLength[v_] := 
      Sqrt[v[[1]]*v[[1]] + v[[2]]*v[[2]] + 
          v[[3]]*v[[3]]]

VectorLength[n1]

VectorLength2[v_] := 
      v[[1]]*v[[1]] + v[[2]]*v[[2]] + 
        v[[3]]*v[[3]]

len1  =  VectorLength2[n1];
len2  =  VectorLength2[n2];
len3  =  VectorLength2[n3];
If[len1 == 0,
If[
        len2 == 
          0, 
{* n1 and n2 are zero length *}
, 
{* n1 is zero length,  
          n2 is not *}
],
If[
        len2 == 
          0, 
{* n1 is not zero lenght,  
          n2 is zero length *}
, 
{* neither n1 nor n2 is zero length *}
If[
            VectorLength2[n1] < VectorLength2[n2], 
If[
              VectorLength2[n1] < 
                VecgtorLength2[n3], 
axis = 
                CrossProduct[n2, n3], 
axis = 
                CrossProduct[n1, n2]], 
If[
              VectorLength2[n2] < 
                VectorLength2[n3], 
axis = 
                CrossProduct[n1, n3], 
axis = 
                CrossProduct[n1, 
                  n2]]
]
]

n1

n2

n3

Clear[f1]

f1

f2