MHT CET · Maths · Three Dimensional Geometry
The vector equation of the plane passing through the point \(\mathrm{A}(1,2,-1)\) and parallel to the vectors \(2 \hat{i}+\hat{j}-\hat{k}\) and \(\hat{i}-\hat{j}+3 \hat{k}\) is
- A \(\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=-9\)
- B \(\overline{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})=9\)
- C \(\overline{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})=9\)
- D \(\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=-9\)
Answer & Solution
Correct Answer
(D) \(\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=-9\)
Step-by-step Solution
Detailed explanation
Let \(\left(x_1, y_1, z_1\right)=(1,2,-1)\), \(\mathrm{a}_1, \mathrm{~b}_1, \mathrm{c}_1=2,1,-1\) and : \(\mathrm{a}_2, \mathrm{~b}_2, \mathrm{c}_2=1,-1,3\)
\(\therefore \quad\) the equation of required plane is
\(\begin{aligned}
& \left|\begin{array}{ccc}
x-x_1 & y-y_1 & z-z_1 \\
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2
\end{array}\right|=0 \\
& \Rightarrow\left|\begin{array}{ccc}
x-1 & y-2 & z+1 \\
2 & 1 & -1 \\
1 & -1 & 3
\end{array}\right|=0 \\
& \Rightarrow 2 x-2-7 y+14-3 z-3=0 \\
& \Rightarrow 2 x-7 y-3 z+9=0 \\
& \Rightarrow \bar{r} \cdot(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=-9
\end{aligned}\)
\(\therefore \quad\) the equation of required plane is
\(\begin{aligned}
& \left|\begin{array}{ccc}
x-x_1 & y-y_1 & z-z_1 \\
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2
\end{array}\right|=0 \\
& \Rightarrow\left|\begin{array}{ccc}
x-1 & y-2 & z+1 \\
2 & 1 & -1 \\
1 & -1 & 3
\end{array}\right|=0 \\
& \Rightarrow 2 x-2-7 y+14-3 z-3=0 \\
& \Rightarrow 2 x-7 y-3 z+9=0 \\
& \Rightarrow \bar{r} \cdot(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=-9
\end{aligned}\)
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