$$left{eginarraylcccc mboxLine 1 & : & x = 1 + 2a, & y = 2 - a, & z = 4 - 2a\<1mm>mboxLine 2 & : & !! x = 9 + b, & ,,, y = 5 + 3b,& ,,z = -4-bendarrayest.$$Point of Intersection: $left<7,-1,-2 ight>$.

You are watching: Find parametric equations for the line through that is perpendicular to the plane

How to find parametric equation of the line which is perpendicular to these $2$ lines and passes though suggest of interarea ?.


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Hint:

The coefficient vector of the parameter $a$ represents the "direction vector" of that line, and taking the cross product with one more vector will provide a perpendicular direction.

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Take a airplane $alpha$ passing through the allude and also perpendicular to line 1.Take a plane $eta$ passing throught the allude and also perpendicular to line 2.The interarea $alpha cap eta$ is the line you"re trying to find.


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$ ewcommandbx<1>,box<15px,border:1px groove navy>displaystyle#1, ewcommandraces<1>leftlbrace,#1, ight brace ewcommandracks<1>leftlbrack,#1, ight brack ewcommanddd eurosoup.orgrmd ewcommandds<1>displaystyle#1 ewcommandexpo<1>, eurosoup.orgrme^#1, ewcommandic eurosoup.orgrmi ewcommandmc<1> eurosoup.orgcal#1 ewcommandmrm<1> eurosoup.orgrm#1 ewcommandpars<1>left(,#1, ight) ewcommandpartiald<3><>fracpartial^#1 #2partial #3^#1 ewcommand oot<2><>,sqrt<#1>,#2,, ewcommand otald<3><>frac eurosoup.orgrmd^#1 #2 eurosoup.orgrmd #3^#1 ewcommandverts<1>leftvert,#1, ightvert$eginalign&left{eginarraylclmboxLine 1 & : & vecr =overbracephantomAvecr_1phantomA^dsparseginarrayc1 \ 2 \ 4endarray + a\overbracephantomAvecn_1phantomA^dsparseginarrayr2 \ -1 \ -2endarray\<1mm>--- & -- & -------------\<1mm>mboxLine 2 & : & vecr =underbracephantomAvecr_2phantomA_dsparseginarrayr9 \ 5 \ -4endarray + a\underbracephantomAvecn_2phantomA_dsparseginarrayr1 \ 3 \ -1endarrayendarrayappropriate.endalign

The answer is given byeginalignmbox &\box<10px,#ffd,border:1px groove navy>%parseginarraycx \ y \ zendarray & =parseginarrayr7 \ -1 \ -2endarray +a,vecn_1 imes vecn_2 =parseginarrayr7 \ -1 \ -2endarray +avertseginarrayrhatx & haty & hatz\2 & -1 & 2\1 & 3 & -1endarray =parseginarrayr7 \ -1 \ -2endarray +aparseginarrayr-1 \ 4 \ 7endarray\<5mm> & =box<10px,#ffd,border:1px groove navy>%parseginarrayrcr7 & - & a\ -1 & + &4a \ -2 & + & 7aendarray\ &endalign