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Question
Show that the line `bar"r" = (2hat"j" - 3hat"k") + lambda(hat"i" + 2hat"j" + 3hat"k") and bar"r" = (2hat"i" + 6hat"j" + 3hat"k") + mu(2hat"i" + 3hat"j" + 4hat"k")` are coplanar. Find the equation of the plane determined by them.
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Solution
The lines `bar"r" = bar"a"_1 + lambda_1bar"b"_1 and bar"r" = bar"a"_2 + lambda_2bar"b"_2` are coplanar If `bar"a"_1.(bar"b"_1 xx bar"b"_2) = bar"a"_2.(bar"b"_1 xx bar"b"_2)`
Here `bar"a"_1 = 2hat"j" - 3hat"k", bar"a"_2 = 2hat"i" + 6hat"j" + 3hat"k"`,
`bar"b"_1 = hat"i" + 2hat"j" + 3hat"k", bar"b"_2 = 2hat"i" + 3hat"j" + 4hat"k"`
∴ `bar"a"_2 - bar"a"_1 = (2hat"i" + 6hat"j" + 3hat"k") - (2hat"j" - 3hat"k")`
= `2hat"i" + 4hat"j" + 6hat"k"`
`bar"b"_1 xx bar"b"_2 = |(hat"i" ,hat"j",hat"k"),(1, 2, 3),(2, 3, 4)|`
= `(8 - 9)hat"i" - (4 - 6)hat"j" + (3 - 4)hat"k"`
= `-hat"i" + 2hat"j" - hat"k"`
∴ `bar"a"_1.(bar"b"_1 xx bar"b"_2) = (2hat"j" - 3hat"k").(-hat"i" + 2hat"j" - hat"k")`
= 0(– 1) + 2(2) + (– 3)(– 1)
= 0 + 4 + 3
= 7
and `bar"a"_2.(bar"b"_1 xx bar"b"_2) = (2hat"i" + 6hat"j" + 3hat"k").(-hat"i" + 2hat"j" - hat"k")`
= 2(– 1) + 6(2) + 3(– 1)
= –2 + 12 – 3
= 7
∴ `bar"a"_1.(bar"b"_1 xx bar"b"_2) = bar"a"_2.(bar"b"_1 xx bar"b"_2)`
Hence, the given lines are coplanar.
The plane determined by these lines is given by
∴ `bar"r".(bar"b"_1 xx bar"b"_2) = bar"a"_1.(bar"b"_1 xx bar"b"_2)`
i.e. `bar"r".(-hat"i" + 2hat"j" - hat"k")` = 7
Hence, the given lines are coplnar and the equation of the plane determined bt these lines is
`bar"r".(-hat"i" + 2hat"j" - hat"k")` = 7.
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