First, calculate the vectors \(\overrightarrow{PN}\) and \(\overrightarrow{PM}\).

\(\overrightarrow{PN} = 8\mathbf{i} - 8\mathbf{k}\)

\(\overrightarrow{PM} = 4\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\)

Calculate the dot product \(\overrightarrow{PN} \cdot \overrightarrow{PM}\):

\(\overrightarrow{PN} \cdot \overrightarrow{PM} = (8)(4) + (0)(4) + (-8)(-6) = 32 + 0 + 48 = 80\)

Calculate the magnitudes:

\(|\overrightarrow{PN}| = \sqrt{8^2 + 0^2 + (-8)^2} = \sqrt{128}\)

\(|\overrightarrow{PM}| = \sqrt{4^2 + 4^2 + (-6)^2} = \sqrt{68}\)

Use the cosine formula:

\(\sqrt{128} \times \sqrt{68} \times \cos M \hat{P} N = 80\)

\(\cos M \hat{P} N = \frac{80}{\sqrt{128} \times \sqrt{68}}\)

Calculate \(M \hat{P} N\):

\(M \hat{P} N = \cos^{-1}\left(\frac{80}{\sqrt{128} \times \sqrt{68}}\right) \approx 31.0^{\circ}\) awrt