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Question
The position vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by ______.
Options
`-3hati - 5hatj - 10hatk`
`21hati + 3hatj`
`11hati + 9hatj - 2hatk`
`-11hati - 9hatj + 2hatk`
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Solution
The position vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by `underlinebb(11hati + 9hatj - 2hatk)`.
Explanation:
∵ Diagonals of parallelogram bisect each other.
So, point O is the mid-point of diagonal AC.
Coordinates of O = `((4 + 12)/2, (2 + 4)/2, (-6 + 5)/2)`
= `(8, 3, -1/2)`
O will also be mid-point of BD.
`\implies ((x + 5)/2, (y - 3)/2, (z + 1)/2) = (8, 3, -1/2)`
On comparing both sides, we get
`\implies` x = 11, y = 9, z = – 2
So, position vector of D is `11hati + 9hatj - 2hatk`
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