Show that the sum of these four vectors is $0$.

Question

Four vectors are erected perpendicularly to the four faces of a general tetrahedron. Each vector is pointing outwards and has a length equal to the area of the face. Show that the sum of these four vectors is 0.

Let $A, B,$ and $C$ be vectors representing the three edges starting from a fixed vertex. Express each of the four vectors in terms of $A, B$, and $C$, and show that their sum is the zero vector; do not introduce a coordinate system.

I was trying to: let's call the four vectors $$v_1,v_2,v_3,v_4$$

we know that:

$$\frac{AC \times BC}{2} = v_1$$

I was trying to express the other vectors but I would need a fourth edge to do that using origin as edge we have

$$\frac{OC \times OA}{2} = v_2$$ $$\frac{OB \times OC}{2} = v_3$$ $$\frac{OA \times OB}{2} = v_4$$

hence: $$s =\frac{AC \times BC}{2} + \frac{OC \times OA}{2}+ \frac{OB \times OC}{2} + \frac{OA \times OB}{2}$$

$$s = \frac{1}{2}((AC \times BC) + (C \times A) + (B \times C )+ (A \times B) )$$

$$s = \frac{1}{2}(((C-A) \times (C-B)) + (C \times A) + (B \times C )+ (A \times B) )$$

Answer 1

In these types of problems, you must carefully account for the direction of the vectors. Let's choose $ABC$ in such a way that they form a triangle going clockwise, and $O$ a point above the $ABC$ plane, like in the picture below: We choose the convention that $XY$ is the vector with the origin in $X$ and the tip in $Y$. Then the sum of the areas multiplied by $2$ is $$2S=OA\times OB+OB\times OC+OC\times OA-BC\times BA$$ It does not matter which vertex you choose in the last term (I choose $B$), the important part is to make sure that the cross product points outwards. The way I've made the choice, $BC\times BA$ points up, in the same side of the $ABC$ plane as $O$. That's why I've put a negative sign for the last term.

Now $$OB+BC=OC\\OB+BA=OA$$so $$BC\times BA=(OC-OB)\times(OA-OB)\\=OC\times OA-OB\times OA-OC\times OB+OB\times OB\\=OC\times OA+OA\times OB+OB\times OC$$ I've uses $a\times b=-b\times a$ for the terms with the negative sign, and $a\times a=0$ for the last term.

With this, $$2S=OA\times OB+OB\times OC+OC\times OA-(OC\times OA+OA\times OB+OB\times OC)=0$$