Let the vectors a =- i + j +3 k and b = i +3 j + k. For some λ, μ ∈ R, let c =λ a +μ b. If c ·(3 i -6 j +2 k )=10 and c ·( i + j + k )=-2, then | c…

Let the vectors $\displaystyle \overrightarrow{\mathrm{a}}=-\hat{i}+\hat{j}+3 \hat{k}$ and $\displaystyle \overrightarrow{\mathrm{b}}=\hat{i}+3 \hat{j}+\hat{k}$. For some $\displaystyle \lambda, \mu \in \mathbf{R}$, let $\displaystyle \overrightarrow{\mathrm{c}}=\lambda \overrightarrow{\mathrm{a}}+\mu \overrightarrow{\mathrm{b}}$. If $\displaystyle \vec{c} \cdot(3 \hat{i}-6 \hat{j}+2 \hat{k})=10$ and $\displaystyle \vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})=-2$, then $\displaystyle |\vec{c}|^2$ is equal to: