If the vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are coplanar, then \(\left|\begin{array}{ccc}a & b & c \ a \cdot a & a \cdot b & a \cdot c \ b \cdot a & b \cdot b & b \cdot c\end{array}\right|\) is equal to

Since, \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are coplanar, there must exists three scalars \(x, y\) and \(z\) are not all zero such that
\(
x \mathbf{a}+y \mathbf{b}+z \mathbf{c}=0
\)
Multiplying both sides of Eq. (i) by a and \(\mathbf{b}\) respectively, we get
\(
\begin{array}{l}
x \mathbf{a} \cdot \mathbf{a}+y \mathbf{a} \cdot \mathbf{b}+z \mathbf{a} \cdot \mathbf{c}=0 \\
x \mathbf{b} \cdot \mathbf{a}+y \mathbf{b} \cdot \mathbf{b}+z \mathbf{b} \cdot \mathbf{c}=0
\end{array}
\)
Eliminating \(x, y\) and \(z\) from Eqs. (i), (ii) and
(iii), we get
\(
\left|\begin{array}{ccc}
\mathbf{a} & \mathbf{b} & \mathbf{c} \\
\mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\
\mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c}
\end{array}\right|=0
\)