The two adjacent sides of a parallelogram are `2 hat i-4 hat j+5 hat k` and ` hat i-2 hat j-3 hat k` . Find the unit vector parallel to one of its diagonals. Also, find its area.

Explanation

The adjacent sides of a parallelogram are `vec(a) = 2hat(i) - 4hat(j) + 5hat(k)` and `vec(b) = hat(i) - 2hat(j) - 3hat(k)` then the diagonal of the parallelogram is `vec(x) = vec(a) + vec(b)`.
`implies vec(x) = 2hat(i) - 4hat(j) + 5hat(k) + hat(i) - 2hat(j) - 3hat(k)`
`= 3hat(i) - 6hat(j) + 2hat(k)`
`therefore |vec(x)| = sqrt((3)^(2) + (-6)^(2) + (2)^(2))`
`=sqrt(9+36+4) = sqrt(49)=7`
Then unit vector parallel to the diagonal
`=(vec(x))/(|vec(x)|)=(3hat(i)-6hat(j)+2hat(k))/(7) = (3)/(7)hat(i) - (6)/(7) hat(j) + (2)/(7)hat(k)`
and area of the parallelogram
`=|axxb|=|(hat(i),hat(j),hat(k)),(2,-4,5),(1,-2,-3)|`
`=|hat(i) (12 + 10) - hat(j)(-6 -5) + hat(k)(-4+4)|`
`=|22hat(i) + 11hat(j) + 0hat(k)|`
`=sqrt((22)^(2)+(11)^(2)+0^(2))`
`=11sqrt(5)` sq. units.