Vector Algebra

Let a vector \({\overrightarrow a }\) be coplanar with vectors \(\overrightarrow b = 2\widehat i + \widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j + \widehat k\). If \({\overrightarrow a}\) is perpendicular to \(\overrightarrow d = 3\widehat i + 2\widehat j + 6\widehat k\), and \(\left| {\overrightarrow a } \right| = \sqrt {10}\). Then a possible value of \([\matrix{ {\overrightarrow a } & {\overrightarrow b } & {\overrightarrow c } \cr } ] + [\matrix{ {\overrightarrow a } & {\overrightarrow b } & {\overrightarrow d } \cr } ] + [\matrix{ {\overrightarrow a } & {\overrightarrow c } & {\overrightarrow d } \cr } ]\) is equal to :