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Question
Find vector equation of the plane passing through A(−2 ,7 ,5) and parallel to vectors `4hat"i" - hat"j" + 3hat"k"` and `hat"i" + hat"j" + hat"k"`
Solution
The plane is passing through the point A(−2, 7, 5) and parallel to vectors
`4hat"i" - hat"j" + 3hat"k"` and `hat"i" + hat"j" + hat"k"`
∴ `bar"a" = -2hat"i" + 7hat"j" + 5hat"k"`
Let `bar"b"_1 - 4hat"i" - hat"j" + 3hat"k"` and `bar"b"_2 = hat"i" + hat"j" + hat"k"`
The given plane is perpendicular to the vector
`bar"n" = bar"b"_1 xx bar"b"_2`
= `|(hat"i", bar"j", hat"k"),(4, -1, 3),(1, 1, 1)|`
= `hat"i"(-1 - 3) - hat"j"(4 - 3) + hat"k"(4 + 1)`
`bar"n" = -4hat"i" - hat"j" + 5hat"k"`
Vector equation of a plane is `bar"r"*bar"n" = bar"a"*bar"n"`
∴ `bar"r"*(-4hat"i" - hat"j" + 5hat"k") = (-2hat"i" + 7hat"j" + 5hat"k")*(-4hat"i" - hat"j" + 5hat"k")`
= (−2)(−4) + (7)(−1) + (5)(5)
= 8 − 7 + 25
∴ `bar"r"*(-4hat"i" - hat"j" + 5hat"k")` = 26
i.e., `bar"r"*(4hat"i" + hat"j" - 5hat"k")` = −26
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