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Question
Solve the following :
Find the vector equation of the plane passing through the point A(– 2, 3, 5) and parallel to the vectors `4hat"i" + 3hat"k" and hat"i" + hat"j"`.
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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" + 3hat"j" + 5hat"k"`
`bar"b" = 4hat"i" + 3hat"k"`,
`bar"c" = hat"i" + hat"j"`
∴ `bar"b" xx bar"c" = |(hat"i", hat"j", hat"k"),(4, 0, 3),(1, 1, 0)|`
= `(0 - 3)hat"i" - (0 - 3)hat"j" - (4 - 0)hat"k"`
= `-3hat"i" + 3hat"j" + 4hat"k"`
∴ `bar"r".(bar"b" xx bar"c") = bar"a".(bar"b" xx bar"c")`
`bar"r". (-3hat"i" + 3hat"j" + 4hat"k") = (-2hat"i" + 3hat"j" + 5hat"k").(-3hat"i" + 3hat"j" + 4hat"k")`
= 6 + 9 + 20
= 35
∴ From (1), the vector equation of the required plane is `bar"r".(- 3hat"i" + 3hat"j" + 4hat"k")` = 35.
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