Q63 / 150 · MHT CET 2011 · Mathematics

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

  • A
    1
  • Correct answer
    B
    $\underline{0}$
  • C
    $-1$
  • D
    None of these

Solution

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 $$ \left|\begin{array}{ccc} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} & \mathbf{a} \cdot \mathbf{c} \\ \mathbf{b} \cdot \mathbf{a} & \mathbf{b} \cdot \mathbf{b} & \mathbf{b} \cdot \mathbf{c} \end{array}\right|=0 $$

Hence, the correct answer is option: 'B'.

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