Glenn Hopkins

2022-07-18

We want to find $O\left(x,y,z\right)$ by using two known points $P\left(2,3,5\right)$ and $Q\left(1,2,2\right)$ and angle of $POQ=60$ degrees with plane $E:1.15714286x+1.8547619y-z-2.86101191=0$ and length $OP=OQ$
Is there any way to find it?

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Expert

You have 3 conditions:
1). point $O\left(x,yz\right)$ lies on plane E, so
$1.15714286x+1.8547619y-z=2.86101191$
2). $OP=OQ$
$\left(x-2{\right)}^{2}+\left(y-3{\right)}^{2}+\left(z-5{\right)}^{2}=\left(x-1{\right)}^{2}+\left(y-2{\right)}^{2}+\left(z-2{\right)}^{2}.$
This simplifies to
$2x+2y+6z=29.$
3). $\mathrm{\angle }POQ={60}^{\circ }$ and $OP=OQ$ so $\mathrm{△}POQ$ is equilateral. Thus
$O{Q}^{2}=P{Q}^{2}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\left(x-1{\right)}^{2}+\left(y-2{\right)}^{2}+\left(z-2{\right)}^{2}=11.$
The answer will not be unique.

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