If the vectors $\bar{a}=\hat{\imath}-2 \hat{\jmath}+\hat{k}, \bar{b}=2 \hat{\imath}-5 \hat{\jmath}+p…

If the vectors $\bar{a}=\hat{\imath}-2 \hat{\jmath}+\hat{k}, \bar{b}=2 \hat{\imath}-5 \hat{\jmath}+p \hat{k}$ and $\bar{c}=5 \hat{\imath}-9 \hat{\jmath}+4 \hat{k}$ are coplanar, then the value of $\mathrm{p}$ is
  1. $-3$
  2. 3
  3. $\frac{1}{3}$
  4. $-\frac{1}{3}$

Solution

Given vectors are coplanar $\Rightarrow \bar{a} \cdot(\bar{b} \times \bar{c})=0$ $\begin{aligned} &\left|\begin{array}{rrr} 1 & -2 & 1 \\ 2 & -5 & \mathrm{P} \\ 5 & -9 & 4 \end{array}\right|=0 \\ \therefore & 1(-20+9 \mathrm{P})+2(8-5 \mathrm{P})+1(-18+25)=0 \\ &-20+9 \mathrm{P}+16-10 \mathrm{P}+7=0 \Rightarrow-\mathrm{P}+3=0 \Rightarrow \quad \mathrm{P}=3 \end{aligned}$

Asked in: MHT CET 2020 (13 Oct Shift 1)

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