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Question
Find the parametric form of vector equation and Cartesian equations of straight line passing through (5, 2, 8) and is perpendicular to the straight lines `vec"r" = (hat"i" + hat"j" - hat"k") + "s"(2hat"i" - 2hat"j" + hat"k")` and `vec"r" = (2hat"i" - hat"j" - 3hat"k") + "t"(hat"i" + 2hat"j" + 2hat"k")`
Solution
GIven point `vec"a" = 5hat"i" + 2hat"j" + 8hat"k"`
`vec"b" = 2hat"i" + hat"j" - 2hat"k"`
`vec"d" = hat"i" + 2hat"j" + 2hat"k"`
`vec"b" xx vec"d" = |(hat"i", hat"j", hat"k"),(2, -2, 1),(1, 2, 2)|`
= `- 6hat"i" - 3hat"j" + 6hat"k"`
∴ This’ vector is perpendicular to both the given straight lines.
∴ The required straight line is `vec"r" = vec"a" + "t"(vec"b" xx vec"d")`
`vec"r" = (5hat"i" + 2hat"j" + 8hat"k") + "t"(- 6hat"i" - 3hat"j" + 6hat"k")`
OR
`vec"r" = (5hat"ii" + 2hat"j" + 8hat"") + "t"(2hat"i" + hat"j" - 2hat"k"), "t" ∈ "R"`
`(x - 5)/(-6) = (y - 2)/(-3) = (z - 8)/6`
`(x - 5)/2 = (y - 2)/1 = (z - 8)/(-2)`
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