= Solution
A <quadratic form> on a finite-dimensional real <vector space> $V$ is a function $q(v)=B(v,v)$ for some symmetric <bilinear form> $B$. A <positive-definite quadratic form> satisfies $q(v)>0$ for every $v\ne0$.
Completing squares gives
$$
x^2+2xy+2y^2+2yz+3z^2=(x+y)^2+(y+z)^2+2z^2.
$$
Set $u=x+y$, $v=y+z$, $w=z$. The corresponding ordered basis in the original coordinates is
$$
\boxed{(1,0,0),\ (-1,1,0),\ (1,-1,1),}
$$
and the diagonal form is $u^2+v^2+2w^2$. All diagonal coefficients are positive, so \b[the form is positive definite].
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