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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.
Solution
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
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