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Question
Show that the points `(hat"i" - hat"j" + 3hat"k")` and `3(hat"i" + hat"j" + hat"k")` are equidistant from the plane `vec"r" * (5hat"i" + 2hat"j" - 7hat"k") + 9` = 0 and lies on opposite side of it.
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Solution
Given points are `"P"(hat"i" - hat"j" + 3hat"k")` and `"Q"3(hat"i" + hat"j" + hat"k")` and the plane `vec"r" * (5hat"i" + 2hat"j" - 7hat"k") + 9` = 0
Perpendicular distance of `"P"(hat"i" - hat"j" + 3hat"k")` from the plane
`vec"r" * (5hat"i" + 2hat"j" - 7hat"k") + 9 = |((hat"i" - hat"j" + 3hat"k")*(5hat"i" + 2hat"j" - 7hat"k") + 9)/sqrt((5)^2 + (2)^2 + (-7)^2)|`
= `|(5 - 2 - 21 + 9)/sqrt(25 + 4 + 49)|`
= `|(-9)/sqrt(78)|`
And perpendicular distance of `"Q"(3hat"i" + 3hat"j" + 3hat"k")` from the plane
= `|((3hat"i" + 3hat"j" + 3hat"k")*(5hat"i" + 2hat"j" - 7hat"k") + 9)/sqrt(25 + 4 + 29)|`
= `|(15 + 6 - 21 + 9)/sqrt(78)|`
= `|9/sqrt(78)|`
Hence, the two points are equidistant from the given plane.
Opposite sign shows that they lie on either side of the plane.
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