Why doesn't the proving work if the vectors change sides?

Question

Instead of calculating the area of F4 as shown in the solution I could also calculate it by doing 1/2 (B-A)x(C-A) since for area the direction doesn't matter but if i do it this way the proving doesn't seem to work? I dont understand why? The direction of the vector isn't supposed to effect the area so i could do it either way and still the proving should work.

Answer 1

Let $\mathbf V_1,$ $\mathbf V_2,$ $\mathbf V_3,$ $\mathbf V_4$ be vectors whose magnitudes are respectively equal to the areas of $F_1,$ $F_2,$ $F_3,$ $F_4$ and whose directions are perpendicular to those faces in the outward direction.

I have added emphasis to the words "in the outward direction."

When you write $\frac12 (\mathbf B-\mathbf A)\times(\mathbf C-\mathbf A)$ you get a vector that is perpendicular to the bottom face of the tetrahedron but is pointing in the inward direction (to the same side of the face as the side where the rest of the tetrahedron lies) using the right-hand rule for the direction of the cross product. So this vector does not satisfy the conditions in the statement, which say every vector must point in the outward direction from its face.