1.

Find thecoordinates offoot ofperpendicularand thelengthofthe perpendiculardrawnfrom thepointP(5,4,2)to the linethis line .

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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