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Question
Show that the straight lines `vec"r" = (5hat"i" + 7hat"j" - 3hat"k") + "s"(4hat"i" + 4hat"j" - 5hat"k")` and `vec"r"(8hat"i" + 4hat"j" + 5hat"k") + "t"(7hat"i" + hat"j" + 3hat"k")` are coplanar. Find the vector equation of the plane in which they lie
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Solution
Let `vec"a" = 5hat"i" + 7hat"j" - 3hat"k"`
`vec"b" = 4hat"i" + 4hat"j" - 5hat"k"`
`vec"c" = 8hat"i" + 4hat"j" + 5hat"k"`
`vec"d" = 7hat"i" + hat"j" + 3hat"k"`
We know that given two lines are coplanar if
`(vec"c" - vec"a")*(vec"b" xx vec"d")` = 0 ......(1)
`vec"b" xx vec"d" = |(vec"i", vec"j", vec"k"),(4, 4, -5),(7, 1, 3)|`
= `vec"i"(12 + 5) - vec"j"(12 + 35) + vec"k"(4 - 28)`
`vec"b" xx vec"d" = 17hat"i" - 47hat"j" - 24hat"k"`
`vec"c" - vec"a" = (8hat"i" + 4hat"j" + 5hat"k") - (5hat"i" + 7hat"j" - 3hat"k") = 3hat"i" - 3hat"j" + 8hat"k"`
(1) ⇒ `(3hat"i" - 3hat"j" + 8hat"k")*(17hat"i" - 47hat"j" - 24hat"k")` = 51 + 141 – 192 = 0
∴ The two given lines are colpanar so, the non-parametric vector equation is
`(vec"r" - vec"a")*(vec"b" xx vec"d")` = 0
`vec"r"*(vec"b" xx vec"d") = vec"a"*(vec"b" xx vec"d")`
`vec"r"*(17vec"i" - 47vec"j" - 24vec"k") = (5vec"i" + 7vec"j" - 3vec"k")(17vec"i" - 47vec"j" - 24vec"k")`
`vec"r"*(17vec"i" - 47vec"j" - 24vec"k")` = 85 – 329 + 72
⇒ `vec"r"*(17vec"i" - 47vec"j" - 24vec"k")` = – 172
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