<s>[INST] Geometrical meaning of $x^2+y^2+z^2-xy-xz-yz$ I am looking for a geometrical interpretation of the symmetrical expression
$$f=x^2+y^2+z^2-xy-xz-yz\tag{1}$$
with $x,y,z \in \mathbb{R}$. I could for example $f$ interprete as dot products of a vector with its permuted vector
$$f=\begin{pmatrix}x\\y\\z\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}-\begin{pmatrix}x\\y\\z\end{pmatrix}\begin{pmatrix}y\\z\\x\end{pmatrix}\tag{2}$$
however I think there are more symmetrical ways to represent $f$ geometrically. Maybe $x,y,z$ can be thought of sides of a triangle, etc. Geometrical interpretations in $\mathbb{R},\mathbb{R^2}, \mathbb{R^3}$ are of interest.
 [/INST] Render $(x-y)^2=x^2-2xy+y^2$ and similarly for the $x,z$ and $y,z$ pairs. When you add up the squares you get
$2(x^2+y^2+z^2-xy-xz-yz).$
So $x^2+y^2+z^2-xy-xz-yz$ is half the sum of squared distances between three points on a line.
</s>