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Question
Show that the points `(hati - hatj + 3hatk)` and `3(hati + hatj + hatk)` are equidistant from the plane `vecr * (5hati + 2hatj - 7hatk) + 9` = 0 and lies on opposite side of it.
Solution
To show that these given points `(hati - hatj + 3hatk)` and `3(hati + hatj + hatk)` are equidistant from the plane `vecr * (5hati + 2hatj - 7hatk) + 9` = 0
And are equidastant from the plane,
We have to prove that midpoint of these points lies on the plane.
Now midpoint of the given points is `2hati + hatj + 3hatk`
On substituting vector(r) by the mid point in plane, we get
L.H.S. = `(2hati + hatj + 3hatk) * (5hati + 2hatj - 7hatk) + 9`
= 10 + 2 – 21 + 9
= 0
= R.H.S.
Hence, the two points lie on opposite sides of the plane are equidistant form the plane.
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