[Math] Finding normal vectors of polygons

Question

I have the following diagram:

I want to find the normal vector for the polygon of points $abc$ and the plane highlighted in red with the points $bcde$.

To find the normal vector for the polygon of the the points $abc$, what I did was to find the vector $\vec{ac}=oc-oa$ and $\vec{ab}=ob-oa$, where $o$ is the origin, and cross them together. So it turns out to be $ac \times ab = \begin{bmatrix}
2\\
-2\\
2
\end{bmatrix}
\times
\begin{bmatrix}
3\\
-3\\
-3
\end{bmatrix}=
\begin{bmatrix}
12\\ 12
\\
0
\end{bmatrix}$.

And for the plane highlighted in red wit hthe points $bcde$, I used the same method by finding any two vectors, $\vec{ed}$ and $\vec{ab}$ and cross them together. So, $\vec{ed} \times \vec{ab}=\begin{bmatrix}
0\\
0\\
2
\end{bmatrix}
\times
\begin{bmatrix}
3\\
-3\\
-3
\end{bmatrix}=
\begin{bmatrix}
6\\ 6
\\
0
\end{bmatrix}$.

But then now, by looking at the picture, how could the 2 planes have the same normal vector when they are so off in their own direction? What have I done wrong? Is what I have done finding the right normal vectors in the first place?

Answer 1

The vector $\vec{ab}$ does not lie in the "red plane" defined by bcde. Try instead: $\vec{ed} \times \vec{dc}$

Also, be careful about whether your normals point inward or outward. The first vector you computed points somewhat upward (out of the polygon).