The order of $[xyz]\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ is:

Let the matrices be $A$, $B$, and $C$.The order of matrix $A=[xyz]$ is $1\times 3$.The order of matrix $B=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$ is $3\times 3$.The order of matrix $C=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ is $3\times 1$.Calculating the product $A\times B$:
$$(1\times 3)\cdot (3\times 3)=1\times 3$$
Calculating the final product $(AB)\times C$:
$$(1\times 3)\cdot (3\times 1)=1\times 1$$