Let a =9 i -13 j +25 k, b =3 i +7 j -13 k and c =17 i -2 j + k be three given vectors. If r is a vector such that r × a =( b + c ) × a and r ·( b - c…

Let $\displaystyle \vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}$ and $\displaystyle \vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}$ be three given vectors. If $\displaystyle \vec{r}$ is a vector such that $\displaystyle \vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \times \vec{a}$ and $\displaystyle \vec{r} \cdot(\vec{b}-\vec{c})=0$, then $\displaystyle \frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2}$ is equal to $\displaystyle \_\_\_\_$.