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

[tengxiaona]

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

Homework Statement

(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.

Homework Equations

projection formula, corss product

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)

......

......

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 ,please give me a hand

thanks!

 

[tengxiaona]

i get some ideas from another topic

https://www.physicsforums.com/showthread.php?t=274638"

i use the equation wut Dick offered X_p(s)-X_q(t)=u*W

then i get

(4,4,4)+t[-3,-6,-9] - (3,-1,5)+s[0,0,-5] = u[30,-15,0]<--cross product by deriction vectors

i solve it and get t=11/15 s=38/25 u=-1/25

wuts next?

PLZ HELP

 

[tengxiaona]

any1 help me ! please!

 

[Billy Bob]

Your original posted solution to (a) gives the dist from C to K, but how do you know C is the closest pt to K? Your original (a) is wrong because K and L don't lie in the same plane.

Look at your second post.

The skew lines K and L lie on two parallel planes P1 and P2. The distance between K and L is the distance between the two planes P1 and P2.

Your vector [30,-15,0] is perpendicular to both planes P1 and P2.

(4,4,4)+t[-3,-6,-9] when t=11/15 is pt M on line K

(3,-1,5)+s[0,0,-5] when s=38/25 is pt N on line L

Vector MN is parallel to [30,-15,0].

You are ready to write down the answer to both parts now, do you see it?