For two lines to be perpendicular, the product of their slopes must be -1. Let's first find the slope of the line joining the points (2, P) and (-1, 3). The slope is given by:

\(m_1 = \frac{3 - P}{-1 - 2} = \frac{P - 3}{3}\)

Now, let's find the slope of the line joining the points (P, 4) and (6, -2). The slope is given by:

\(m_2 = \frac{-2 - 4}{6 - P} = \frac{-6}{P - 6}\)

Since these two lines are perpendicular, we have:

\(m_1 \cdot m_2 = -1\)

\(\frac{P - 3}{3} \cdot \frac{-6}{P - 6} = -1\)

\(\frac{2(P - 3)}{P - 6} = -1\)

Multiplying both sides by \(P - 6\) gives:

\(2P - 6 = -P + 6\)

\(3P = 12\)

\(P = 4\)