Let λ ∈ Z, a =λ i + j - k and b =3 i - j +2 k. Let c be a vector such that ( a + b + c ) × c = 0, a · c =-17 and b · c =-20. Then | c ×(λ i + j + k…

Let $\displaystyle \lambda \in \mathbb{Z}, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\displaystyle \vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. Let $\displaystyle \vec{c}$ be a vector such that $\displaystyle (\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=-17$ and $\displaystyle \vec{b} \cdot \vec{c}=-20$. Then $\displaystyle |\vec{c} \times(\lambda \hat{i}+\hat{j}+\hat{k})|^2$ is equal to