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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry1 Mark
Question
State True or False for the following: The angle between the line $\vec{\text{r}}=(5\hat{\text{i}}-\hat{\text{j}}-4\hat{\text{k}})+\lambda(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}})$ and the plane $\vec{\text{r}}(3\hat{\text{i}}-4\hat{\text{j}}-\hat{\text{k}})+5=0$ is $\sin^{-1}\Big(\frac{5}{2\sqrt{91}}\Big).$
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Answer

False.Solution:
We have, $\vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$ and $\vec{\text{n}}=3\hat{\text{i}}-4\hat{\text{j}}-\hat{\text{k}}$
Let $\theta$ is the angle between line and plane.
Then, $\sin\theta=\frac{|\vec{\text{b}}\cdot\vec{\text{n}}|}{|\vec{\text{b}}|\cdot|\vec{\text{n}}|}$
$=\bigg|\frac{(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k})}\cdot(3\hat{\text{i}}-4\hat{\text{j}}-\hat{\text{k}})}{\sqrt{6}\cdot\sqrt{26}}\bigg|$
$=\frac{|6+4-1|}{\sqrt{156}}=\frac{9}{2\sqrt{39}}$
$\therefore\theta=\sin^{-1}\frac{9}{2\sqrt{39}}$

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