Let α ∈ R and the three vectors a =α i + j +3 k, b =2 i + j -α k and c =α i -2 j +3 k. Then the set S ={α: a, b and c are coplanar }

Let $\displaystyle \alpha \in \mathbf{R}$ and the three vectors $$\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}+3 \hat{k}, \quad \overrightarrow{\mathrm{~b}}=2 \hat{i}+\hat{j}-\alpha \hat{k} $$ and $\displaystyle \overrightarrow{\mathrm{c}}=\alpha \hat{i}-2 \hat{j}+3 \hat{k}$. Then the set $\displaystyle \mathrm{S}=\{\alpha: \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ are coplanar $\displaystyle \}$