Q265 / 270 · NDA 2020 · Mathematics

If the vectors $\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{c}=\hat{j}+p \hat{k}$ are coplanar, then what is the value of $p$ ?

  • A
    1
  • Correct answer
    B
    -1
  • C
    5
  • D
    -5

Solution

As the given vectors are coplanar, then $\vec{c} \times \vec{a} \times \vec{b}=0 \Rightarrow\left|\begin{array}{ccc} 0 & 1 & p \\ 2 & -3 & 1 \\ 1 & 2 & -3 \end{array}\right|=0$ $\begin{aligned} & \Rightarrow 0+1(1+6)+p(4+3)=0 \\ & \Rightarrow 7+7 p=0 \Rightarrow p=-1\end{aligned}$

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

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