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Question
Find the non-parametric form of vector equation and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to thestraight lines `(x - 1)/2 = (y + 1)/3 = (x - 3)/1` and `(x + 3)/2 = (y - 3)/(-5) = (z + 1)/(-3)`
Solution
`vec"a" = 2hat"i" + 3hat"j" + 6hat"k"`
`vec"b" = 2hat"i" + 3hat"j" + hat"k"`
`vec"c" = 2hat"i" - 5hat"j" - 3hat"k"`
Non-parametric form of vector equation
`(vec"r" - vec"a")*(vec"b" xx vec"c")` = 0
`vec"b" xx vec"c" = |(hat"i", hat"j", hat"k"),(2, 3, 1),(2, -5, -3)|`
= `hat"i"(- 9 + 5) -hat"j"(- 6 - 2) + hat"k"(- 10 - 6)`
= `- 4hat"i" + 8hat"j" - 16hat"k"`
= `-4(hat"i" - 2hat"j" + 4hat"k")`
`[vec"r" - (2hat"i" + 3hat"j" + 6hat"k")]*[hat"i" + 2hat"j" + 4hat"k"]` = 0
`vec"r"*(hat"i" - 2hat"j" + 4hat"k") = 2 - 6 + 24`
`vec"r"*(hat"i" - 2hat"j" + 4hat"k")` = 20
Cartesian equation
`(xhat"i" + yhat"j" + zhat"k")*(hat"i" - 2hat"j" + 4hat"k")` = 20
`x - 2y + 4z - 20` = 0
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