The plane ell x+my=0 is rotated about its lin | vedclass

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(A) Let the equation of the plane after rotation be $P_{3}: \ell x+my+nz=0$.The line of intersection of the planes $P_{1}: \ell x+my=0$ and $P_{2}: z=

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$\ell x+my \pm z 	an \alpha \sqrt{\ell^{2}+m^{2}}=0$

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(A) Let the equation of the plane after rotation be $P_{3}: \ell x+my+nz=0$.The line of intersection of the planes $P_{1}: \ell x+my=0$ and $P_{2}: z=0$ is the line where $\ell x+my=0$ and $z=0$.The normal vectors are $\vec{n}_{1} = (\ell, m, 0)$ and $\vec{n}_{3} = (\ell, m, n)$.The angle $\alpha$ between the planes $P_{1}$ and $P_{3}$ is given by $\cos \alpha = \frac{|\vec{n}_{1} \cdot \vec{n}_{3}|}{|\vec{n}_{1}| |\vec{n}_{3}|}$.$\cos \alpha = \frac{|\ell^{2}+m^{2}|}{\sqrt{\ell^{2}+m^{2}} \sqrt{\ell^{2}+m^{2}+n^{2}}} = \sqrt{\frac{\ell^{2}+m^{2}}{\ell^{2}+m^{2}+n^{2}}}$.Squaring both sides, $\cos^{2} \alpha = \frac{\ell^{2}+m^{2}}{\ell^{2}+m^{2}+n^{2}}$.$\Rightarrow \cos^{2} \alpha (\ell^{2}+m^{2}+n^{2}) = \ell^{2}+m^{2}$.$\Rightarrow n^{2} \cos^{2} \alpha = (\ell^{2}+m^{2})(1 - \cos^{2} \alpha) = (\ell^{2}+m^{2}) \sin^{2} \alpha$.$\Rightarrow n^{2} = (\ell^{2}+m^{2}) 	an^{2} \alpha$.$\Rightarrow n = \pm \sqrt{\ell^{2}+m^{2}} 	an \alpha$.Substituting $n$ into the equation of $P_{3}$, we get $\ell x+my \pm z \sqrt{\ell^{2}+m^{2}} 	an \alpha = 0$.

The plane $\ell x+my=0$ is rotated about its line of intersection with the plane $z=0$ through an angle $\alpha$. The equation of the new plane is

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