1. Find the magnitude of $\vec A$

Given, $\vec A = 3\hat i + 4\hat j$

Its magnitude is $|\vec A| = \sqrt{3^2+4^2} = \sqrt{9+16} = 5$

So, the required vector must also have magnitude $5$.

2. Use the direction of $\vec B$

Given, $\vec B = 4\hat i + 3\hat j$

A vector parallel to $\vec B$ must be of the form $\vec V = k(4\hat i + 3\hat j)$

Since it lies in the first quadrant, both components are positive, so $k>0$.

3. Match the magnitude

Magnitude of $\vec B$ is $|\vec B| = \sqrt{4^2+3^2} = 5$

Thus, $|\vec V| = |k|\,|\vec B| = 5|k|$

But the required magnitude is $5$, so $5|k| = 5 \implies |k|=1$

Since the vector is in the first quadrant, $k=1$.

Hence, $\vec V = 4\hat i + 3\hat j$

4. Read off the components

The vector has components $(x,3)$.

Comparing with $\vec V = (4,3)$, $x=4$

Final Answer

$\boxed{4}$

The derived answer matches the stored correct answer.