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प्रश्न
Lines `overliner = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk)` and `overliner = (4hati - 3hatj + 2hatk) + μ(hati - 2hatj + 2hatk)` are coplanar. Find the equation of the plane determined by them.
उत्तर
Lines `overliner = overlinea_1 + λ_1overlineb_1` and `overliner = overlinea_2 + λ_2overlineb_2` are coplanar if and only if `(overlinea_2 - overlinea_1) . (overlineb_1 xx overlineb_2)` = 0
Here `overlinea_1 = hati + hatj - hatk, overlinea_2 = 4hati - 3hatj + 2hatk`
`overlineb_1 = 2hati - 2hatj + hatk, overlineb_2 = hati - 2hatj + 2hatk`
∴ `overlinea_2 - overlinea_1 = 3hati - 4hatj + 3hatk`
`(overlinea_2 - overlinea_1) . (overlineb_1 xx overlineb_2) = |(3, -4, 3),(2, -2, 1),(1, -2, 2)|`
= 3(–2) + 4(3) + 3(–2)
= –6 + 12 – 6
= 0
∴ Given lines are coplanar.
Now `overlineb_1 xx overlineb_2 = |(hati, hatj, hatk),(2, -2, 1),(1, -2, 2)|`
`overlineb_1 xx overlineb_2 = -2hati - 3hatj - 2hatk`
The equation of the plane determined by them is `(overliner - overlinea_1).(overlineb_1 xx overlineb_2)` = 0
∴ `overliner.(overlineb_1 xx overlineb_2) = overlinea_1.(overlineb_1 xx overlineb_2)`
∴ `overliner.(-2hati - 3hatj - 2hatk) = (hati + hatj - hatk).(-2hati - 3hatj - 2hatk)`
∴ `overliner.(-2hati - 3hatj - 2hatk)` = – 3
∴ `overliner.(2hati + 3hatj + 2hatk)` = 3
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