Let a vector a be coplanar with vectors b =2 i + j + k and c = i - j + k. If a is perpendicular to d =3 i +2 j +6 k, and | a |=√ 10. Then a possible…

Let a vector $\displaystyle \vec{a}$ be coplanar with vectors $\displaystyle \vec{b}=2 \hat{i}+\hat{j}+\hat{k}$ and $\displaystyle \vec{c}=\hat{i}-\hat{j}+\hat{k}$. If $\displaystyle \vec{a}$ is perpendicular to $\displaystyle \overrightarrow{\mathrm{d}}=3 \hat{i}+2 \hat{j}+6 \hat{k}$, and $\displaystyle |\overrightarrow{\mathrm{a}}|=\sqrt{10}$. Then a possible value of $\displaystyle \left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right]+\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{d}\end{array}\right]+\left[\begin{array}{lll}\vec{a} & \vec{c} & \vec{d}\end{array}\right]$ is equal to :