(i) To find \(a\), we use the similarity of triangles. The triangle \(OPD\) is similar to \(OPA\). The ratio of the sides gives:

\(\frac{10 - a}{10} = \frac{6}{10}\)

Solving for \(a\):

\(10 - a = 6\)

\(a = 4\)

(ii) To express \(\overrightarrow{BG}\), we find the coordinates of \(B\) and \(G\). \(B\) is at \((10, 10, 0)\) and \(G\) is at \((0, 6, 4)\). Thus,

\(\overrightarrow{BG} = (0 - 10)\mathbf{i} + (6 - 10)\mathbf{j} + (4 - 0)\mathbf{k}\)

\(\overrightarrow{BG} = -10\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}\)

(iii) To find angle \(GBA\), we use the scalar product:

\(\overrightarrow{BG} \cdot \overrightarrow{BA} = 40\)

\(\cos GBA = \frac{40}{\sqrt{132} \times \sqrt{100}}\)

\(GBA = \cos^{-1}\left(\frac{40}{\sqrt{13200}}\right)\)

\(GBA = 69.6^\circ\)