The sides of a parallelogram are `2hati +4hatj -5hatk and hati + 2hatj +3hatk `. The unit vector parallel to one of the diagonals is

Explanation

Let OABC be the given parallelogramm and let the adjacent sides OA and OB be represented by `a=2hati+4hatj-5hatk and b=hati+2hatj+3hatk` respectively.
Now, the vectors along the two diagonals are

The required unit vectors are
`hatnn_(1)=(d)/(|d_(1)|)=(3hati+6hatj-2hatk)/(sqrt(3^(2)+6^(2)+(-2)^(2)))`
`=(3)/(7)hati+(6)/(7)hatj-(2)/(7)hatk`
and `hatn_(2)=(d_(2))/(|d_(2)|)=(-hati-2hatj+8hatk)/(sqrt((-1)^(2)+(-2)^(2)+8^(2)))`
`=(-1)/(sqrt(69))hati-(2)/(sqrt(69))hatj+(8)/(sqrt(69))hatk`