IIT-JEE 1989Question 1
If vectors $\vec A,\vec B,\vec C$ are coplanar, show that $$\begin{vmatrix}\vec a & \vec b & \vec c\\ \vec a\cdot\vec a & \vec a\cdot\vec b & \vec a\cdot\vec c\\ \vec b\cdot\vec a & \vec b\cdot\vec b & \vec b\cdot\vec c\end{vmatrix}=\vec 0$$
Vector Algebra
Answerproved: determinant = zero vector
Solution
Expanding along row 1: $\vec aM_1-\vec bM_2+\vec cM_3$ with scalar $2\times2$ minors. Coplanar: $\vec c=s\vec a+t\vec b$, so in the scalar rows column 3 $=s\,$col1$+t\,$col2. With $D=$ det(col1,col2): $M_1=-sD$, $M_2=tD$, $M_3=D$. Sum $=D(\vec c-s\vec a-t\vec b)=\vec 0$.