Question

If a, b, c are coplanar vectors, then

• A. $\left| \begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{b} & \mathbf{c} & \mathbf{a} \\ \mathbf{c} & \mathbf{a} & \mathbf{b} \end{matrix} \right| = \mathbf{0}$
• B. $\left| \begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a}.\mathbf{a} & \mathbf{a}.\mathbf{b} & \mathbf{a}.\mathbf{c} \\ \mathbf{b}.\mathbf{a} & \mathbf{b}.\mathbf{b} & \mathbf{b}.\mathbf{c} \end{matrix} \right| = \mathbf{0}$
• C. $\left| \begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{c}.\mathbf{a} & \mathbf{c}.\mathbf{b} & \mathbf{c}.\mathbf{c} \\ \mathbf{b}.\mathbf{a} & \mathbf{b}.\mathbf{c} & \mathbf{b}.\mathbf{b} \end{matrix} \right| = \mathbf{0}$
• D. $\left| \begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a}.\mathbf{b} & \mathbf{a}.\mathbf{a} & \mathbf{a}.\mathbf{c} \\ \mathbf{c}.\mathbf{a} & \mathbf{c}.\mathbf{c} & \mathbf{c}.\mathbf{b} \end{matrix} \right| = \mathbf{0}$

Answer 1

Answer: (B)

$\left| \begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a}.\mathbf{a} & \mathbf{a}.\mathbf{b} & \mathbf{a}.\mathbf{c} \\ \mathbf{b}.\mathbf{a} & \mathbf{b}.\mathbf{b} & \mathbf{b}.\mathbf{c} \end{matrix} \right| = \mathbf{0}$

Answer 2

Since $\mathbf{a},\mathbf{b}$ and $\mathbf{c}$ are coplanar, therefore there exists $(x,y,z$not all zero) such that

$x\mathbf{a} + y\mathbf{b} + z\mathbf{c} = 0$ .....(i)

Multiply be $\mathbf{a}$ scalarly, we get

$x(\mathbf{a}.\mathbf{a}) + (\mathbf{a}.\mathbf{b}) + z(\mathbf{a}.\mathbf{c}) = 0$ ......(ii)

and $x(\mathbf{a}.\mathbf{b}) + y(\mathbf{b}.\mathbf{b}) + z(\mathbf{b}.\mathbf{c}) = 0$ .....(iii)

Eliminating $x,y$ and $z$ from (i), (ii) and (iii),

we get $\left| \begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a}.\mathbf{a} & \mathbf{a}.\mathbf{b} & \mathbf{a}.\mathbf{c} \\ \mathbf{a}.\mathbf{b} & \mathbf{b}.\mathbf{b} & \mathbf{b}.\mathbf{c} \end{matrix} \right| = 0$.

Note: Students should remember this question as a formula.