||The 3x3 determinant trick is just a special case for when you want to calculate a cross-product in R**3. The actual definition says nothing about a determinant -- it merely states that
|[A, B]| is equal to
|A| |B| sin(A^B), and that the resulting vector points in a direction perpendicular to both
B, according to the "right hand rule".
From this definition we can deduce that for two linearly-dependent vectors the cross-product is equal to zero. (We can think of this fact as just a part of the definition, but who cares...)
We know from linear algebra that two vectors are linearly dependent if the determinant of their respective column-matrix is equal to zero. Since in our case the respective matrix isn't even square we can be sure that the matrix is linearly dependent. (But note that you don't need to calculate determinants to figure out that two vectors in R**1 are linearly dependent; that's pretty obvious anyways, isn't it?)
What you did was to project vectors in R**1 onto vectors in R**3 and then calculated the respective cross-product -- which is not in fact justified! First of all, the way you simply picked any old arbitrary way of projecting vectors onto a basis of a different dimension was not very honest -- stricly speaking, the two vectors are in different linear spaces, and may or may not be related by weird correlations. Second of all, you still have to prove that linear dependence in R**1 implies linear dependence in R**3.
All in all, I think you need to get rid of the bad habit of using ready-made algorithms in math. But, that seems to be the global disease of the American education system...
Peace and much love...