If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are unit vectors and \(\theta\) is angle between \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{c}}\) and \(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}+2 \overline{\mathrm{c}}=\overline{0}\), then \(|\overline{\mathrm{a}} \times \overline{\mathrm{c}}|=\)

\(
\begin{aligned}
& \overline{\mathrm{a}}+2 \overline{\mathrm{b}}+2 \overline{\mathrm{c}}=\overline{0} \\
& \Rightarrow \mathrm{a}+2 \overline{\mathrm{c}}=-2 \overline{\mathrm{b}}
\end{aligned}
\)
Squaring on both sides, we get
\(
\begin{aligned}
& |\overline{\mathrm{a}}|^2+4 \overline{\mathrm{a}} \cdot \overline{\mathrm{c}}+4|\overline{\mathrm{c}}|^2=4|\overline{\mathrm{b}}|^2 \\
& \Rightarrow 1+4|\overline{\mathrm{a}}||\overrightarrow{\mathrm{c}}| \cos \theta+4=4 \\
& \Rightarrow \cos \theta=-\frac{1}{4} \\
& \Rightarrow \sin \theta=\frac{\sqrt{15}}{4} \\
& |\overline{\mathrm{a}} \times \overline{\mathrm{c}}|=|\overline{\mathrm{a}}||\overline{\mathrm{c}}| \sin \theta \\
& \quad=(1)(1)\left(\frac{\sqrt{15}}{4}\right)=\frac{\sqrt{15}}{4}
\end{aligned}
\)