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Question
Find the distance between the lines:
`vecr = (hati + 2hatj - 4hatk) + λ(2hati + 3hatj + 6hatk)`;
`vecr = (3hati + 3hatj - 5hatk) + μ(4hati + 6hatj + 12hatk)`
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Solution
Lines are parallel
`a_1/a_2 = b_1/b_2 = c_1/c_2`
`2/4 = 3/6 = 6/12`
`1/2 = 1/2 = 1/2`
`vecb = 2hati + 3hatj + 6hatk`
`|vecb| = sqrt(4 + 9 + 36)`
= `sqrt(49)`
= 7
S.D. = `|(vecb xx (veca_2 - veca_1))/|vecb||`
`veca_1 = hati + 2hatj - 4hatk`
`veca_2 = 3hati + 3hatj - 5hatk`
`veca_2 - veca_1 = 2hati + hatj - hatk`
`vecb xx (veca_2 - veca_1) = |(hati, hatj, hatk),(2, 3, 6),(2, 1, -1)|`
= `-9hati + 14hatj - 4hatk`
`|vecb xx (veca_2 - veca_1)| = sqrt(81 + 196 + 16)`
= `sqrt(293)`
S.D. = `sqrt(293)/7` units.
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