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`

Answer 1

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`