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The plane — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThe plane3 Marks
Question
Determine the value of $\lambda$ for which the following planes are perpendicular to other.
$\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=7$ and $\vec{\text{r}}\cdot(\lambda\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})=26$
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Answer

We know that the planes $\vec{\text{r}}\cdot\vec{\text{n}}_1=\text{d}_1,\vec{\text{ r}}\cdot\vec{\text{n}}_2=\text{d}_2$ are perpendicular to each other only if $\vec{\text{n}}_1\cdot\vec{\text{n}}_2=0$
Here, $\vec{\text{n}}_1=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}, \vec{\text{n}}_2=\lambda\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}}$
The given planes are perpendicular.
$\Rightarrow\vec{\text{n}}_1\cdot\vec{\text{n}}_2=0$
$\Rightarrow(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})\cdot(\lambda\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})=0$
$\Rightarrow\lambda+4-21=0$
$\Rightarrow\lambda-17=0$
$\Rightarrow\lambda=17$

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