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The plane — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsThe plane5 Marks
Question
Find the vector equation of the following planes in non-parametric form.
$\vec{\text{r}}=(2\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})+\lambda(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})+\mu(5\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}})$
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Answer

We know that equation $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}+\mu\vec{\text{c}} $ represents a plane passing through a point whose position vector is $\vec{\text{a}}$ and parallel to t.Here, $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\hat{\text{c}}=5\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}$
Normal vector, $\vec{\text{n}}=\vec{\text{b}}\times\vec{\text{c}}$
$=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\1&2&3\\5&-2&7\end{vmatrix}$
$=20\hat{\text{i}}+8\hat{\text{j}}-12\hat{\text{k}}$
The vector equation of the plane in scalar product form is,
$\vec{\text{r}}\cdot\vec{\text{n}}=\vec{\text{a}}\cdot\vec{\text{n}}$
$\Rightarrow\vec{\text{r}}\cdot(20\hat{\text{i}}+8\hat{\text{j}}-12\hat{\text{k}})=(2\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})(20\hat{\text{i}}+8\hat{\text{j}}-12\hat{\text{k}})$
$\Rightarrow\vec{\text{r}}\cdot\big(4(5\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})\big)=40+16+12$
$\Rightarrow\vec{\text{r}}\cdot\big(4(5\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})\big)=68$
$\Rightarrow\vec{\text{r}}\cdot(5\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})=17$

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