Let a =2 i +5 j - k, b =2 i -2 j +2 k and c be three vectors such that ( c + i ) ×( a + b + i )= a ×( c + i ). If a · c =-29, then c ·(-2 i + j + k )…

Let $\displaystyle \overrightarrow{\mathrm{a}}=2 \hat{i}+5 \hat{j}-\hat{k}, \overrightarrow{\mathrm{~b}}=2 \hat{i}-2 \hat{j}+2 \hat{k}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ be three vectors such that $\displaystyle (\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i})=\vec{a} \times(\vec{c}+\hat{i})$. If $\displaystyle \vec{a} \cdot \vec{c}=-29$, then $\displaystyle \vec{c} \cdot(-2 \hat{i}+\hat{j}+\hat{k})$ is equal to: