Let a =2 i -3 j +4 k, b =3 i +4 j -5 k and a vector c be such that a ×( b + c )+ b × c = i +8 j +13 k. If a · c =13, then (24- b · c ) is equal to…

Let $\displaystyle \vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k}$ and a vector $\displaystyle \vec{c}$ be such that $\displaystyle \vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}$. If $\displaystyle \vec{a} \cdot \vec{c}=13$, then $\displaystyle (24-\vec{b} \cdot \vec{c})$ is equal to $\displaystyle \_\_\_\_$.