Let c be the projection vector of b =λ i +4 k, λ>0, on the vector a = i +2 j +2 k. If | a + c |=7, then the area of the parallelogram formed by the…

Let $\displaystyle \vec{c}$ be the projection vector of $\displaystyle \vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0$, on the vector $\displaystyle \vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $\displaystyle |\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\displaystyle \vec{b}$ and $\displaystyle \vec{c}$ is $\displaystyle \_\_\_\_$.