Mathematics Question from JEE Main 2025 · 22 January, Shift 1

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