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Question
Find the vector equation of the plane passing through the point A(– 2, 7, 5) and parallel to vector `4hat"i" - hat"j" + 3hat"k" and hat"i" + hat"j" + hat"k"`.
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Solution
The vector equation of the plane passing through the point A`(bara)` and parallel to the vectors `bar"b" and bar"c"` is
`bar"r".(bar"b" xx bar"c") = bar"a".(bar"b" xx bar"c")` ...(1)
Here, `bar"a" = -2hat"i" + 7hat"j" + 5hat"k"`
`bar"b" = 4hat"i" - hat"j" + 3hat"k"`,
`bar"c" = hat"i" + hat"j" + hat"k"`
∴ `bar"b" xx bar"c" = |(hat"i", hat"j", hat"k"),(4, -1, 3),(1, 1, 1)|`
= `(-1 - 3)hat"i" - (4 - 3)hat"j" + (4 - (-1))hat"k"`
= `-4hat"i" - hat"j" + 5hat"k"`
∴ `bar"a".(bar"b" xx bar"c") = (-2hat"i" + 7hat"j" +5hat"k").(-4hat"i" - hat"j" + 5hat"k")`
= (– 2)(– 4) + (7)(–1) + (5)(5)
= 8 – 7 + 25
= 1 + 25
= 26
∴ From (1), the vector equation of the required plane is `bar"r".(- 4hat"i" - hat"j" + 5hat"k")` = 26.
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