If A=(1,3,-5) and B=(3,5,-3), then the vector equation of the plane passing through the midpoint of AB and perpendicular to AB is

r . (^i +^j +^k) = 2

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r . (^i +^j -^k) = 2

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r . (^i -^j +^k) = 2

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Solution

The correct option is A $r . (^i +^j +^k) = 2$
$- - \to A B = - - \to O B - - - \to O A = (3^i + 5^j - 3^k) - (^i + 3^j - 5^k) = 2^i + 2^j + 2^k$
Midpoint of AB is (2, 4, -4)
Vector equation of the plane is $[r - (2^i + 4^j - 4^k)] . (2^i + 2^j + 2^k) = 0$
$\Rightarrow r . (^i +^j +^k) = 2 + 4 - 4 \Rightarrow r . (^i +^j +^k) = 2$

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If A=(1,3,-5) and B=(3,5,-3), then the vector equation of the plane passing through the midpoint of AB and perpendicular to AB is

The correct option is A r.(^i+^j+^k)=2−−→AB=−−→OB−−−→OA=(3^i+5^j−3^k)−(^i+3^j−5^k)=2^i+2^j+2^k Midpoint of AB is (2, 4, -4) Vector equation of the plane is [r−(2^i+4^j−4^k)].(2^i+2^j+2^k)=0 ⇒r.(^i+^j+^k)=2+4−4⇒r.(^i+^j+^k)=2