JEE Advance - Mathematics (1989 - No. 5)
If vectors $$\overrightarrow A ,\overrightarrow B ,\overrightarrow C $$ are coplanar, show that
$$$\left| {\matrix{
{} & {\overrightarrow {a.} } & {} & {\overrightarrow {b.} } & {} & {\overrightarrow {c.} } \cr
{\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {c.} } \cr
{\overrightarrow {b.} } & {\overrightarrow {a.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {b.} } & {\overrightarrow {c.} } \cr
} } \right| = \overrightarrow 0 $$$
Since the vectors are coplanar, their scalar triple product is zero.
The given determinant represents the scalar triple product of the vectors.
Expanding the determinant will always result in a non-zero value.
If vectors are coplanar, then they are linearly independent.
The determinant is zero because the rows are linearly dependent.