10. Given that the circumcenter of $\triangle A B C$ is $O$, and $2 \overrightarrow{O A}+3 \overrightarrow{O B}+4 \overrightarrow{O C}=\mathbf{0}$, then the value of $\cos \angle B A C$ is $\qquad$
$$
\begin{array}{l}
3 \overrightarrow{O B}+4 \overrightarrow{O C}=-2 \overrightarrow{O A} \Rightarrow 9+16+24 \cos \angle B O C=4 \Rightarrow \cos \angle B O C=-\frac{7}{8} \\
\Rightarrow 2 \cos ^{2} \angle B A C-1=-\frac{7}{8} \Rightarrow \cos \angle B A C=\frac{1}{4} .
\end{array}
$$