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The vector equation of the plane passing through the point $\mathrm{A}(1,2,-1)$ and parallel to the vectors…
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
$\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})=-9$ $\overline{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})=9$ $\overline{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}})=9$ $\overline{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=-9$ Solution
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
$\begin{aligned}
& \left|\begin{array}{ccc}
x-x_1 & y-y_1 & z-z_1 \\
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2
\end{array}\right|=0 \\
& \Rightarrow\left|\begin{array}{ccc}
x-1 & y-2 & z+1 \\
2 & 1 & -1 \\
1 & -1 & 3
\end{array}\right|=0 \\
& \Rightarrow 2 x-2-7 y+14-3 z-3=0 \\
& \Rightarrow 2 x-7 y-3 z+9=0 \\
& \Rightarrow \bar{r} \cdot(2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=-9
\end{aligned}$
Asked in: MHT CET 2024 (10 May Shift 2)
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