Answer
$\theta=23.4^{\circ}$
Work Step by Step
Use the $x, y, z$ axes.
$
\begin{aligned}
\mathbf{u}_{A B} & =\left(\frac{-0.5 \mathbf{i}+0.2 \mathbf{j}+0.2 \mathbf{k}}{0.57446}\right) \\
& =-0.8704 \mathbf{i}+0.3482 \mathbf{j}+0.3482 \mathbf{k} \\
\mathbf{F} & =-100 \cos 10^{\circ} \mathbf{i}+100 \sin 10^{\circ} \mathbf{k} \\
\theta & =\cos ^{-1}\left(\frac{\mathbf{F} \cdot \mathbf{u}_{A B}}{F u_{A B}}\right) \\
& =\cos ^{-1}\left(\frac{-100\left(\cos 10^{\circ}\right)(-0.8704)+0+100 \sin 10^{\circ}(0.3482)}{100(1)}\right) \\
& =\cos ^{-1}(0.9176)=23.4^{\circ}
\end{aligned}
$
