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Three Dimensional Geometry — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSThree Dimensional Geometry1 Mark
Question
State True or False for the following:
The angle between the planes $\vec{\text{r}}\cdot(2\hat{\text{i}}-3\hat{\text{j}}+\text{k})=1$ and $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}})=4$ is $\cos^{-1}\Big(\frac{-5}{\sqrt{58}}\Big).$
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Answer

False.Solution:
The angle between two planes $\vec{\text{r}}.\vec{\text{n}}_1=\text{d}_1$ and $\vec{\text{r}}.\vec{\text{n}}_2=\text{d}_2$ is given by $\cos\theta=\frac{|\vec{\text{n}}_1\cdot\vec{\text{n}}_2|}{|\vec{\text{n}}_1||\vec{\text{n}}_2|}$
Comparing $\vec{\text{r}}\cdot(2\hat{\text{i}}-3\hat{\text{j}}+\text{k})=1$ and $\vec{\text{r}}\cdot(\hat{\text{i}}-\hat{\text{j}})=4$ with $\vec{\text{r}}.\vec{\text{n}}_1=\text{d}_1$ and $\vec{\text{r}}.\vec{\text{n}}_2=\text{d}_2,$ we get
$\vec{\text{n}}_1=(2\hat{\text{i}}-3\hat{\text{j}}+\text{k})$ and $\vec{\text{n}}_2=(\hat{\text{i}}-\hat{\text{j}})$
Substituting $\vec{\text{n}}_1=(2\hat{\text{i}}-3\hat{\text{j}}+\text{k})$ and $\vec{\text{n}}_2=(\hat{\text{i}}-\hat{\text{j}})$ in $\cos\theta=\frac{|\vec{\text{n}}_1\cdot\vec{\text{n}}_2|}{|\vec{\text{n}}_1||\vec{\text{n}}_2|},$
$\cos\theta=\frac{\big|(2\hat{\text{i}}-3\hat{\text{j}}+\text{k})(\hat{\text{i}}-\hat{\text{j}})\big|}{\sqrt{4+9+1\sqrt{1+1}}}$
$\Rightarrow\cos\theta=\frac{|2+3|}{\sqrt{14}\cdot\sqrt{2}}=\frac{5}{2\sqrt{7}}$
$\Rightarrow\theta=\cos^{-1}\Big(\frac{5}{2\sqrt{7}}\Big)$

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