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Question
Find the parametric vector, non-parametric vector and Cartesian form of the equation of the plane passing through the point (3, 6, – 2), (– 1, – 2, 6) and (6, 4, – 2)
Solution
`vec"a" = 3hat"i" + 6hat"j" - 2hat"k"`
`vec"b" = -hat"i" - 2hat"j" + 6hat"k"`
`vec"b" - vec"a" = -4hat"i" - 8hat"j" + 8hat"k"`
`vec"c" - vec"a" = 3hat"i" - 2hat"j"`
`(vec"b" - vec"a") xx (vec"c" - vec"a") = |(hat"i", hat"j", hat"k"),(-4, -8, 8),(3, -2, 0)|`
= `hat"i"(0 + 16) - hat"j"(0 - 24) + hat"k"(8 + 24)`
= `16hat"i" + 24hat"j" + 32hat"k"`
Parametric equation:
`vec"r" = vec"a" + "s"(vec"b" - vec"a") + "t"(vec"c" - vec"a")`
`vec"r" = (3hat"i" + 6hat"j" - 2hat"k") + "s"(-4hat"i" - 8hat"j" + 8hat"k") + "t"(3hat"i" - 2hat"j"), "s", "t" ∈ "R"`
Non-parametric equation:
`(vec"r" - vec"a")*[(vec"b" - vec"a") xx (vec"c" - vec"a")] = vec0`
`(vec"r" - vec"a")*(16hat"i" + 24hat"j" + 32hat"k") = vec0`
`[vec"r"*(16hat"i" + 24hat"j" + 32hat"k")] = (3hat"i" + 6hat"j" - 2hat"k")*(16hat"i" + 24hat"j" + 32hat"k")`
`vec"r"(16hat"i" + 24hat"j" + 32hat"k")` = 128
Cartesian equation:
⇒ `vec"r"(2hat"i" + 3hat"j" + 4hat"k")` = 16
⇒ 2x + 3y + 4z – 16 = 0
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