(i) The position vectors are \(\mathbf{OA} = \mathbf{i} + 7\mathbf{j} + 2\mathbf{k}\) and \(\mathbf{OB} = -5\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\).
The dot product \(\mathbf{OA} \cdot \mathbf{OB} = (1)(-5) + (7)(5) + (2)(6) = -5 + 35 + 12 = 42\).
The magnitudes are \(|\mathbf{OA}| = \sqrt{1^2 + 7^2 + 2^2} = \sqrt{54}\) and \(|\mathbf{OB}| = \sqrt{(-5)^2 + 5^2 + 6^2} = \sqrt{86}\).
Using the dot product formula: \(\mathbf{OA} \cdot \mathbf{OB} = |\mathbf{OA}| |\mathbf{OB}| \cos \theta\), we have:
\(42 = \sqrt{54} \sqrt{86} \cos \theta\).
Solving for \(\cos \theta\), we get \(\cos \theta = \frac{42}{\sqrt{54} \sqrt{86}}\).
Thus, \(\theta = \cos^{-1}\left(\frac{42}{\sqrt{54} \sqrt{86}}\right) \approx 0.907\) radians.
(ii) Given \(\overrightarrow{AB} = 2\overrightarrow{BC}\), we find \(\overrightarrow{BC} = \frac{1}{2} \overrightarrow{AB}\).
\(\overrightarrow{AB} = \mathbf{OB} - \mathbf{OA} = (-5\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}) - (\mathbf{i} + 7\mathbf{j} + 2\mathbf{k}) = -6\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}\).
Thus, \(\overrightarrow{BC} = \frac{1}{2}(-6\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}) = -3\mathbf{i} - \mathbf{j} + 2\mathbf{k}\).
\(\overrightarrow{OC} = \mathbf{OB} + \overrightarrow{BC} = (-5\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}) + (-3\mathbf{i} - \mathbf{j} + 2\mathbf{k}) = -8\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}\).
The magnitude of \(\overrightarrow{OC}\) is \(\sqrt{(-8)^2 + 4^2 + 8^2} = \sqrt{144} = 12\).
The unit vector in the direction of \(\overrightarrow{OC}\) is \(\frac{-8\mathbf{i} + 4\mathbf{j} + 8\mathbf{k}}{12}\).
