(adsbygoogle = window.adsbygoogle || []).push({}); Linear algebar!!!distance between two lines,help for part(b)

1. The problem statement, all variables and given/known data

(a)use cross product to find the distance d from the line K determined by the two points(4,4,4) and (-1,-2,-5) to the line L determined by the parametric equations x=3,y=-1,z=5-5t

(b)find two points P and Q on the lines K and L respectively in part(a) such that the idstance from P to Q is d.

2. Relevant equations

projection formula, corss product

3. The attempt at a solution

i slove part(a)

i set A as (4,4,4) and B as (1,-2,-5), then we can get -->AB=(-3,-6,-9)

from x=3,y=-1,z=5-5t , we get C as (3,-1,-5),then -->AC=(-1,-5,1)

and we know l(distance between L and K)=area of parallelogram/length of AB

which is

l= ||-->AB*-->AC|| / ||-->AB||

now i use cross product -->AB*-->AC=(-3,-6,-9)*(-1,-5,1)=(-39,-12,21)

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FINALLY i get l= sqrt(117/7)

i think i did right way for part(a)

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but for part(b)

i find the projection of -->AC first

proj(AC)={AC*AB/AB*AB}AB

=(3+30-9)/126 (-3,-6,-9)

=4/21(-3,-6,-9)

ok, till here,

can i use (4,4,4)as point P

then use C(3,-1,-5) minus (4/21)(-3,-6,-9) to get point Q?

if i did wrong way ,plz give me a hand

thanks!!!

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# Linear algebar distance between two lines,help for part(b)

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