- #1
tengxiaona
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Linear algebar!distance between two lines,help for part(b)
(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.
projection formula, corss product
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)
-------------------------------------------------------------------------------------
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!
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)
-------------------------------------------------------------------------------------
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!