(i) Since \(CP = \frac{3}{5} CA\), and \(CA = 4\mathbf{i} - 3\mathbf{k}\), we have:

\(\overrightarrow{CP} = \frac{3}{5}(4\mathbf{i} - 3\mathbf{k}) = 2.4\mathbf{i} - 1.8\mathbf{k}\).

(ii) \(\overrightarrow{OP} = \overrightarrow{OA} + \overrightarrow{AP}\). Since \(AP = \frac{3}{5} \times 4\mathbf{i} = 2.4\mathbf{i}\),

\(\overrightarrow{OP} = 2.4\mathbf{i} + 1.2\mathbf{k}\).

\(\overrightarrow{BP} = \overrightarrow{OP} - \overrightarrow{OB} = (2.4\mathbf{i} + 1.2\mathbf{k}) - 2.4\mathbf{j} = 2.4\mathbf{i} - 2.4\mathbf{j} + 1.2\mathbf{k}\).

(iii) Use the scalar product formula:

\(\overrightarrow{BP} \cdot \overrightarrow{CP} = 2.4 \times 2.4 + (-2.4) \times 0 + 1.2 \times (-1.8) = 5.76 - 2.16 = 3.6\).

\(|\overrightarrow{BP}| = \sqrt{2.4^2 + (-2.4)^2 + 1.2^2} = \sqrt{12.96}\),

\(|\overrightarrow{CP}| = \sqrt{2.4^2 + (-1.8)^2} = \sqrt{9}\).

\(\cos BPC = \frac{3.6}{\sqrt{12.96} \times \sqrt{9}} = \frac{1}{3}\).

Angle BPC = \(\cos^{-1}(\frac{1}{3}) \approx 70.5^{\circ}\) (or 1.23 radians).