Given that \(3x + 4y + 6 = 0\) and \(4x - by + 3 = 0\) are perpendicular, find the value of b.
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, \(4x - by + 3 = 0 \implies by = 4x + 3\)
\(y = \frac{4}{b}x + \frac{3}{b}\)
\(\frac{\mathrm d y}{\mathrm d x} = \frac{4}{b}\)
\(\frac{-3}{4} \times \frac{4}{b} = -1 \implies \frac{4}{b} = \frac{4}{3}\)
\(b = 3\)
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