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Question Number 103945 by bobhans last updated on 18/Jul/20

$$acubeABCD.EFGHwithlengthside$$$$4cm.GivenpointPismidpointEF.$$$$findthedistanceoflineAPtoline$$$$HB.$$

Answered by bramlex last updated on 18/Jul/20

$$let:ax+by+cz=-1bea$$$$equationofplaneHBP{}^{\prime }where$$$$BP{}^{\prime }paralleltoAP.$$$$coordinatesA(4,0,0);B(4,4,0)$$$$;P{}^{\prime }(4,6,4)andH(0,0,4)$$$$\left(i\right)substituteB\Rightarrow 4a+4b=-1$$$$\left(ii\right)substituteH\Rightarrow 4c=-1\Rightarrow c=-\frac{1}{4}$$$$\left(iii\right)substituteP$$$${}^{\prime }\Rightarrow 4a+6b+4c=-1\Rightarrow 4a+6b=0$$$$substract\left(i\right)\mathrm{\&}\left(iii\right)$$$$weget2b=1\Rightarrow b=\frac{1}{2}\mathrm{\&}a=-\frac{3}{4}$$$$soeqofplaneHBP{}^{\prime }\Rightarrow -\frac{3}{4}x+\frac{1}{2}y-\frac{1}{4}z+1=0$$$$or-3x+2y-z+4=0$$$$sothedistancewedesiredis\frac{\mid -12+4\mid }{\sqrt{9+4+1}}$$$$=\frac{8}{\sqrt{14}}=\frac{4\sqrt{14}}{7}cm.◼$$

Commented by bramlex last updated on 18/Jul/20

Commented by bobhans last updated on 18/Jul/20

$$waw...niceandcooll$$

Answered by mr W last updated on 18/Jul/20

Commented by mr W last updated on 18/Jul/20

$$A(4,0,4)$$$$P(4,2,0)$$$$H(0,0,0)$$$$B(4,4,4)$$$$eqn.ofAP=(4,0,4)+\lambda (0,1,-2)$$$$eqn.ofHB=(0,0,0)+\mu (1,1,1)$$$$apointonAPisS(4,\lambda ,4-2\lambda )$$$$apointonHBisT(\mu ,\mu ,\mu )$$$$let\mathrm{\Phi }={ST}^2$$$$\mathrm{\Phi }={(4-\mu )}^2+{(\lambda -\mu )}^2+{(4-2\lambda -\mu )}^2$$$$\frac{\partial \mathrm{\Phi }}{\partial s}=2(\lambda -\mu )+4(2\lambda +\mu -4)=0$$$$\Rightarrow 5\lambda +\mu =8...\left(i\right)$$$$\frac{\partial \mathrm{\Phi }}{\partial t}=-2(4-\mu )-2(\lambda -\mu )+2(2\lambda +\mu -4)=0$$$$\Rightarrow \lambda +3\mu =8...\left(ii\right)$$$$\Rightarrow \lambda =\frac{8}{7}$$$$\Rightarrow \mu =\frac{16}{7}$$$${\mathrm{\Phi }}_{min}={(4-\frac{16}{7})}^2+{(\frac{8}{7}-\frac{16}{7})}^2+{(4-\frac{16}{7}-\frac{16}{7})}^2=\frac{224}{49}$$$$min.distancefromAPtoHBis$$$${ST}_{min}=\sqrt{{\mathrm{\Phi }}_{min}}=\sqrt{\frac{224}{49}}=\frac{4\sqrt{14}}{7}$$

Commented by bobhans last updated on 18/Jul/20

$$whatmethoditissir?$$

Commented by bobhans last updated on 18/Jul/20

$$wherebecomes(0,1,-2)sir$$

Commented by mr W last updated on 18/Jul/20

$$\overrightarrow{AP}=(4-4,2-0,0-4)=(0,2,-4)or$$$$(0,1,-2)$$

Commented by bramlex last updated on 18/Jul/20

$$waw...greatyourmethodsir.$$

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