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Question
An insect is crawling along the line `barr = 6hati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)` and another insect is crawling along the line `barr = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)`. At what points on the lines should they reach so that the distance between them s the shortest? Find the shortest possible distance between them.
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Solution
The given lines are non-parallel lines. There is a unique line segment PQ (P lying on one and Q on the other) at right angles to both lines. PQ is the shortest distance between the lines. Hence, the shortest possible distance between the insects = PQ
The position vector of P lying on the line
`barr = 6hati + 2hatj + 2hatk + λ(hati - 2hatj + 2hatk)` is `(6 + λ)hati + (2 - 2λ)hatj + (2 + 2λ)hatk` for some λ
The position vector of Q lying on the line
`vecr = - 4hati - hatk + μ(3hati - 2hatj - 2hatk)` is `(-4 + 3μ)hati + (-2μ)hatj + (-1 - 2μ)hatk` for some μ
`vec(PQ) = (- 10 + 3μ - λ)hati + (-2μ - 2 + 2λ)hatj + (-3 - 2μ - 2λ)hatk`
Since PQ is perpendicular to both lines
`(-10 + 3μ - λ) + (-2μ - 2 + 2λ)(-2) + (-3 - 2μ - 2λ)2` = 0,
i.e., μ – 3λ = 4 ...(i)
And (–10 + 3μ – λ)3 + (–2μ –2 + 2λ)(–2) + (–3 – 2μ – 2λ)(–2) = 0,
i.e., 17μ – 3λ = 20 ...(ii)
Solving (i) and (ii) for λ and μ, we get = 1, 1 = –1.
The position vector of the points, at which they should be so that the distance between them is the shortest, is `5hati + 4hatj` and `-hati - 2hatj - 3hatk`
`vec(PQ) = - 6hati - 6hatj - 3hatk`
The shortest distance = `|vec(PQ)|`
= `sqrt(6^2 + 6^2 + 3^2)`
= 9
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