2023-02-21

Let $\stackrel{\to }{A}$ be vector parallel to line of intersection of planes ${P}_{1}$ and ${P}_{2}$ through origin. ${P}_{1}$ is parallel to the vectors $2\stackrel{^}{j}+3\stackrel{^}{k}$ and $4\stackrel{^}{j}-3\stackrel{^}{k}$ and ${P}_{2}$ is parallel to $\stackrel{^}{j}-\stackrel{^}{k}$ and $3\stackrel{^}{i}+3\stackrel{^}{j}$ then the angle between vector $\stackrel{\to }{A}$ and $2\stackrel{\to }{i}+\stackrel{\to }{j}-2\stackrel{^}{k}$

Jakob Howell

Let vector $\stackrel{\to }{AO}$ be parallel to line of planes ${P}_{1}$ and ${P}_{2}$ through origin.
Normal to plane ${p}_{1}$ is
${\stackrel{\to }{n}}_{1}=\left[\left(2\stackrel{\to }{j}+3\stackrel{\to }{k}\right)×4\stackrel{^}{j}-3\stackrel{^}{k}\right)\right]=-18\stackrel{^}{i}$
Normal to plane ${p}_{2}$ is
$\stackrel{\to }{{n}_{2}}=\left(\stackrel{^}{j}-\stackrel{^}{k}\right)×\left(3\stackrel{^}{i}+3\stackrel{^}{j}\right)=3\stackrel{^}{i}-3\stackrel{^}{j}-3\stackrel{^}{k}$
So, $\stackrel{\to }{OA}$ is parallel to $±\left(\stackrel{\to }{{n}_{1}}×\stackrel{\to }{{n}_{2}}\right)=54\stackrel{^}{j}-54\stackrel{^}{k}$
Angle between $54\left(\stackrel{^}{j}-\stackrel{^}{k}\right)$ and $\left(2\stackrel{^}{i}+\stackrel{^}{j}-2\stackrel{^}{k}\right)$ is
$\mathrm{cos}\theta =±\left(\frac{54+108}{3.54\sqrt{2}}\right)=±\frac{1}{\sqrt{2}}\phantom{\rule{0ex}{0ex}}\theta =\frac{\pi }{4},\frac{3\pi }{4}$

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