Question

The two adjacent sides of a parallelogram are represented by vectors `2hati - 4hatj + 5hatk` and `hati - 2hatj - 3hatk`. Find the unit vector parallel to one of its diagonals, Also, find the area of the parallelogram.

Answer 1

Given two adjacent sides of a parallelogram are `veca = 2hati - 4hatj + 5hatk`

`vecb = hati - 2hatj - 3hatk`

Let `vecc` be the diagonal of given parallelogram.

`vecc = veca + vecb`

= `(2hati - 4hatj + 5hatk) + (hati - 2hatj - 3hatk)`

= `3hati - 6hatj + 2hatk`

∴ `|vecc| = sqrt((3)^2 + (-6)^2 + (2)^2)` = 7

Unit vector in direction of `vecc = vecc/|vecc| = (3hati - 6hatj + 2hatk)/7`

Now, Area of parallelogram = `|veca xx vecb|`

∴ `veca xx vecb = |(hati, hatj, hatk),(2, -4, 5),(1, -2, -3)|`

= `(12 + 10)hati - (-6 - 5)hatj + (-4 + 4)hatk`

= `22hati + 11hatj`

Therefore, Area of parallelogram = `|veca xx vecb|`

= `sqrt((22)^2 + (11)^2)`

= `sqrt(484 + 121)`

= `sqrt(605)` sq.units