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Line and Plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsLine and Plane2 Marks
MCQ
The vector equation of the plane passing through a point having position vector $\hat{ i }-\hat{ j }+\hat{ k }$ and parallel to the vectors $2 \hat{i}+\hat{j}+\hat{k}$ and $\hat{j}+2 \hat{k}$ is
  • A
    $\overline{ r } \cdot(\hat{ i }-4 \hat{ j }-2 \hat{ k })=7$
  • B
    $\overline{ r } \cdot(\hat{ i }+4 \hat{ j }+2 \hat{ k })=7$
  • C
    $\bar{r} \cdot(\hat{i}-4 \hat{j}-2 \hat{k})=-7$
  • ✓
    $\overline{ r } .(\hat{ i }-4 \hat{ j }+2 \hat{ k })=7$
✓

Answer

Correct option: D.
$\overline{ r } .(\hat{ i }-4 \hat{ j }+2 \hat{ k })=7$
(D)
The plane passes through $(1,-1,1)$
This point satisfies the equation of plane in option (D)
Also, it has d.r.s $=\bar{b} \times \bar{c}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & 1 \\ 0 & 1 & 2\end{array}\right|$
$=\hat{ i }(2-1)-\hat{ j }(4-0)+\hat{ k }(2-0)$
$=\hat{ i }-4 \hat{ j }+2 \hat{ k }$
i.e., $1,-4,2$
∴ option (D) is correct answer.
Alternate method:
Let $\bar{a}=\hat{i}-\hat{j}+\hat{k}, \bar{b}=2 \hat{i}+\hat{j}+\hat{k}$ and $\bar{c}=\hat{j}+2 \hat{k}$
Now, $\overline{ b } \times \overline{ c }=\hat{ i }-4 \hat{ j }+2 \hat{ k }$
∴ the vector equation of required plane is $\overline{r} \cdot(\overline{b} \times \overline{c})=\overline{a} \cdot(\overline{b} \times \overline{c}) $
$\Rightarrow \bar{r} \cdot(\hat{i}-4 \hat{j}+2 \hat{k})=(\hat{i}-\hat{j}+\hat{k}) \cdot(\hat{i}-4 \hat{j}+2 \hat{k})$
$\Rightarrow \overline{ r } \cdot(\hat{ i }-4 \hat{ j }+2 \hat{ k })=7$

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