Solution
= Solution
The <dot product> is
$$
\boxed{(a,a+1)\cdot(a+1,a)=2a(a+1)=2a^2+2a.}
$$
Both <vectors> have squared <Euclidean norm> $a^2+(a+1)^2=2a^2+2a+1$. Under the given condition $a^2+a=1/2$, their <dot product> is $1$ and each squared <Euclidean norm> is $2$. Hence
$$
\cos\theta=\frac1{\sqrt2\sqrt2}=\frac12,\qquad\boxed{\theta=\pi/3=60^\circ.}
$$