Two lines are perpendicular to each other if the product of their slopes is -1. The slopes of the lines are given by the coefficients of \(x\) and \(y\) in their equations.

For the first equation \(3y = 4x - 1\), we can rearrange it to solve for \(y\):

\(y = \frac{4}{3}x - \frac{1}{3}\)

So, the slope of the first line is \(m_1 = \frac{4}{3}\).

For the second equation \(Ky = x + 3\), we can rearrange it to solve for \(y\):

\(y = \frac{1}{K}x + \frac{3}{K}\)

So, the slope of the second line is \(m_2 = \frac{1}{K}\).

The two lines are perpendicular, so their slopes should satisfy the condition:

\(m_1 \cdot m_2 = -1\)

\(\frac{4}{3} \cdot \frac{1}{K} = -1\)

Solve for \(K\):

\(\frac{4}{3K} = -1\)

\(4 = -3K\)

\(K = -\frac{4}{3}\)

Therefore, the value of \(K\) is \(-\frac{4}{3}\).