Two lines are given by the equation x = 10 - 8t, y=1+8t, z=15 +2t and x=6-8t, y=2+2t, z=3+6t What is the shortest distance between these two lines?

Ethen Blackwell 2022-07-24 Answered
Two lines are given by the equation x = 10 - 8t, y=1+8t, z=15 +2t and x=6-8t, y=2+2t, z=3+6t What is the shortest distance between these two lines?
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Answers (1)

esbalatzaj
Answered 2022-07-25 Author has 15 answers
any two skew lines lie on parallel planes, and the normal direction of these planes if found using the cross product of the two line directions.
| i j k 8 8 2 8 2 6 | =< 44 , 32 , 48 >
choosing t = 0 on the first line yields the point (10,1,15) and using the normal vector <44,32,48> the plane equation is.
44(x - 10) + 32(y - 1) + 48(z - 15) = 0
44x + 32y + 48z - 1192 = 0 divide by 4.
11x + 8y + 12z - 298 = 0
since the other line is parallel to this plane then the orthogonal distance from any point on that line to this plane is the shortest distance between the two lines, t = 0 yields the point (6,2,3)
Plug into the distance equation from a point to a plane.
D = | ( 11 ) ( 6 ) + ( 8 ) ( 2 ) + ( 12 ) ( 3 ) + ( 298 ) | 11 2 + 8 2 + 12 2 = | 180 | 329 = 180 329 9.92
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