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Question
Find the vector equation of the plane which bisects the segment joining A(2, 3, 6) and B(4, 3, −2) at right angles
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Solution
Let M be the midpoint of segment AB.
A ≡ (2, 3, 6) and B ≡ (4, 3, −2) ......[Given]
By midpoint formula, we get
M ≡ `((x_1+ x_2)/2, (y_1 + y_2)/2, (z_1 +z_2)/2)`
= `((2 + 4)/2, (3 + 3)/2, (6 - 2)/2)`
M ≡ (3, 3, 2)
Since plane bisects the seg AB at right angle,
`bar"n" = bar"AB" = bar"b" - bar"a"`
∴ `bar"n" = 2hat"i" + 0hat"j" - 8hat"k"`
Equation of a plane passing through point M(3, 3, 2) and having a normal `bar"n"` is `bar"r"*bar"n" = bar"m"*bar"n"`
∴ `bar"r"*(2hat"i" - 8hat"k") = (3hat"i" + 3hat"j" + 2hat"k")*(2hat"i" - 8hat"k")`
∴ `bar"r"*(2hat"i" - 8hat"k") = (3)(2) + (3)(0) + (2)(-8)`
∴ `bar"r"*(2hat"i" - 8hat"k")` = – 10
∴ `bar"r"*(hat"i" - hat"k")` = – 5
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