Q78 / 150 · MHT CET 2020 (12 Oct Shift 1) · Mathematics

The value of $m$, if the vectors $\hat{\imath}-\hat{\jmath}-6 \hat{k}, \hat{\imath}-3 \hat{\jmath}+4 \hat{k}$ and $2 \hat{\imath}-5 \hat{\jmath}+m \hat{k}$ are coplanar, is

  • A
    1
  • B
    $-3$
  • Correct answer
    C
    3
  • D
    $-1$

Solution

$\bar{a}=\hat{i}-\hat{j}-6 \hat{k}, \bar{b}=\hat{i}-3 \hat{j}+4 \hat{k}, \bar{c}=2 \hat{i}-5 \hat{j}+m \hat{k}$ are coplanar $\begin{aligned} \therefore &\left|\begin{array}{rrr} 1 & -1 & -6 \\ 1 & -3 & 4 \\ 2 & -5 & \mathrm{~m} \end{array}\right|=0 \\ & 1(-3 \mathrm{~m}+20)+1(\mathrm{~m}-8)-6(-5+6)=0 \\ &-3 \mathrm{~m}+20+\mathrm{m}-8+30-36=0 \Rightarrow-2 \mathrm{~m}+6=0 \quad \Rightarrow \mathrm{m}=3 \end{aligned}$

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

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