InterviewSolution
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                                    Find thecoordinates offoot ofperpendicularand thelengthofthe perpendiculardrawnfrom thepointP(5,4,2)to the linethis line . | 
                            
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Answer» Solution :The vector equation fo thegiven LINE is  `vec( R) =(-hat(i) +3hat(J) +hat(k)) + lambda (2hat(i) +3hat(j) -hat(k))`   Clearlyitpassesthrough the point (-1,3,1) andit hasdirectionrations 2,3,-1 So, itsCartesianequationsare `(x+1)/(2) =(y-3)/(3)=(z-1)/(-1)=r` (say) Thegeneral pointon thisline is (2r -1, 3r+3,-r+1) LetN be thefootof theperpendiculardrawn from thepoint `P(5,4,2)` on the givenline. Thenthis POINTIS `N(2r-6,3r+3-r +1)`forsomefixedvalue of r. D.r' s of PN are(2r-6 ,3r-1 ,-r -1) D.r's ofthegiven lineare 2,3,-1 Since PNis perpendicularto the given line (i) we have `2(2r-6)+3(3r-1)-1,(-r-1) =0 rArr 14r =14 rArr r =1` So , thepointN is givenbyy N(1,6,0) Hencethe foot of theperpendicularfrom thegivenpointP(5,4,2) on thegivenline isN(1,6,0) Let `Q(alpha , beta, gamma)` be theimageof P(5,4,2) in thegiven line . thenN(1,6,0)is themidpoint of PQ. `:.(5+alpha)/(2)=1, (4+beta)/(2) " 6and" (2+gamma)/(2)=0 rArralpha =-3 , beta =8 " and" gamma =-2` Hencethe imageof P(5,4,2)in thegivenlineis Q(-3,8-2)  | 
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