If $\vec{a}= 4\vec{i}-2\hat{j} + 5\hat{k}$ nad $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ represent, both in magnitude and direction, two adjacent sides of a parallelogram, then a ynit vector parallel to the diagonal, which is coinitial with $\vec{a}$ and $\vec{b} $, is :

$\frac{5\hat{i}-4\hat{j}+3\hat{k}}{\sqrt{50}}$

The correct answer is Option (3) → $\frac{5\hat{i}-4\hat{j}+3\hat{k}}{\sqrt{50}}$

Diagonal

$\vec d=\vec a+\vec b=5\hat i-4\hat j+3\hat k$

$\vec v$ such that $\vec v||\vec d$ and $|\vec v|=1$

$\vec v=\frac{\vec d}{|\vec d|}=\frac{5\hat i-4\hat j+3\hat k}{\sqrt{5^2+4^2+3^2}}=\frac{5\hat{i}-4\hat{j}+3\hat{k}}{\sqrt{50}}$