= Solution
For a massive particle on a timelike <world line>, <proper time> satisfies $d\tau=dt\sqrt{1-v^2/c^2}=dt/\gamma$. Its <four-velocity> and <four-acceleration> are $U=dX/d\tau$ and $A=dU/d\tau$. Differentiating $U=\gamma(c,\mathbf v)$ gives
$$
\boxed{A^\mu=\left(\gamma^4\frac{\mathbf v\cdot\mathbf a}{c},\quad
\gamma^2\mathbf a+\gamma^4\frac{\mathbf v\cdot\mathbf a}{c^2}\mathbf v\right).}
$$
The invariant <Minkowski inner products> are
$$
\boxed{U^2=c^2,\qquad P^2=m^2c^2,\qquad P=mU.}
$$
For the explicitly requested contraction,
$$
U\cdot A=\gamma^5\mathbf v\cdot\mathbf a-\gamma^3\mathbf v\cdot\mathbf a
-\gamma^5\frac{v^2}{c^2}\mathbf v\cdot\mathbf a
=\gamma^3\mathbf v\cdot\mathbf a\,[\gamma^2(1-v^2/c^2)-1]=0.
$$
Thus <four-acceleration> is orthogonal to <four-velocity>. Equivalently this follows by differentiating the constant <Lorentz scalar> $U^2=c^2$ with respect to <proper time>.
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