When you have two lines, \(y_{1}, y_{2}\), perpendicular to each other, the product of their slopes = -1.

\(3x + 4y + 6 = 0 \implies 4y = -6 - 3x\)

\(\therefore y = \frac{-6}{4} - \frac{3}{4}x\)

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

Also, \(4x - by + 3 = 0 \implies by = 4x + 3\)

\(y = \frac{4}{b}x + \frac{3}{b}\)

\(\frac{\mathrm d y}{\mathrm d x} = \frac{4}{b}\)

\(\frac{-3}{4} \times \frac{4}{b} = -1 \implies \frac{4}{b} = \frac{4}{3}\)

\(b = 3\)