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Question
Find the parametric form of vector equation, and Cartesian equations of the plane containing the line `vec"r" = (hat"i" - hat"j" + 3hat"k") + "t"(2hat"i" - hat"j" + 4hat"k")` and perpendicular to plane `vec"r"*(hat"i" + 2hat"j" + hat"k")` = 8
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Solution
`vec"a" = hat"i" - hat"j" + 3hat"k"`
`vec"b" = 2hat"i" - hat"j" + 4hat"k"`
`vec"c" = hat"i" + 2hat"j" + hat"k"`
Parametric form of vector equation
`vec"r" = vec"a" + "s"vec"b" + "t"vec"c"`
`vec"r" = (hat"i" - hat"j" + 3hat"k") + "s"(2hat"j" - hat"i" + 4hat"k") + "t"(hat"i" + 2hat"j" + hat"k")`
`vec"b" xx vec"c" = |(hat"i", hat"j", hat"k"),(2, -1, 4),(1, 2, 1)|`
= `hat"i"(- 1 - 8) - hat"j"(2 - 4) + hat"k"(4 + 1)`
= `9hat"i" + 2hat"j" + 5hat"k"`
= `-(9hat"i" - 2hat"j" - 5hat"k")`
Non-parametric form of vector equation:
`(vec"r" - vec"a")*(vec"b" xx vec"c") = vec0`
`[vec"r" - (hat"i" - hat"j" + 3hat"k")]*(-(9hat"i" - 2hat"j" - 5hat"k")) = vec0`
`vec"r"*(+9hat"i" - 2hat"j" - 5hat"k")` = + 9 + 2 – 15
`[vec"r"*(9hat"i" - 2hat"j" - 5hat"k")]` = – 4
Cartesian equation
9x – 2y – 5z = – 4
9x – 2y – 5z + 4 = 0
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