Let a = i -2 j +3 k, b =2 i + j - k, c =λ i + j + k and v = a × b. If v · c =11 and the length of the projection of b on c is p, then 9 p^2 is equal…

Let $\displaystyle \vec{a}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \vec{b}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{c}=\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\displaystyle \vec{v}=\vec{a} \times \vec{b}$. If $\displaystyle \vec{v} \cdot \vec{c}=11$ and the length of the projection of $\displaystyle \vec{b}$ on $\displaystyle \vec{c}$ is $\displaystyle p$, then $\displaystyle 9 p^2$ is equal to