Mathematics Question from JEE Main 2023 · 12 April, Shift 1

Let $\lambda \in \mathbb{Z}$, $\vec{a} = \lambda \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$. Let $\vec{c}$ be a vector such that

$$(\vec{a} + \vec{b} + \vec{c}) \times \vec{c} = 0, \quad \vec{a} \cdot \vec{c} = -17 \quad \text{and} \quad \vec{b} \cdot \vec{c} = -20.$$

Then $\left| \vec{c} \times (\lambda \hat{i} + \hat{j} + \hat{k}) \right|^2$ is equal to: