1. **Problem statement**: Given points A and B with position vectors \(\mathbf{A} = \mathbf{i} + 7\mathbf{j} + 2\mathbf{k}\) and \(\mathbf{B} = -5\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}\), find: (i) the angle \(\angle AOB\) using the scalar product, in radians to 3 significant figures. (ii) point C such that \(\mathbf{AB} = 2\mathbf{BC}\), and find the unit vector in the direction of \(\mathbf{OC}\). 2. **Calculate vector OA and OB**: \[ \mathbf{OA} = \mathbf{i} + 7\mathbf{j} + 2\mathbf{k} \] \[ \mathbf{OB} = -5\mathbf{i} + 5\mathbf{j} + 6\mathbf{k} \] 3. **Calculate the scalar (dot) product \(\mathbf{OA} \cdot \mathbf{OB}\):** \[ (1)(-5) + (7)(5) + (2)(6) = -5 + 35 + 12 = 42 \] 4. **Calculate magnitudes \(|\mathbf{OA}|\) and \(|\mathbf{OB}|\):** \[ |\mathbf{OA}| = \sqrt{1^2 + 7^2 + 2^2} = \sqrt{1 + 49 + 4} = \sqrt{54} = 3\sqrt{6} \approx 7.348 \] \[ |\mathbf{OB}| = \sqrt{(-5)^2 + 5^2 + 6^2} = \sqrt{25 + 25 + 36} = \sqrt{86} \approx 9.274 \] 5. **Calculate the angle between vectors using the dot product formula:** \[ \cos \theta = \frac{\mathbf{OA} \cdot \mathbf{OB}}{|\mathbf{OA}| |\mathbf{OB}|} = \frac{42}{7.348 \times 9.274} = \frac{42}{68.136} \approx 0.6165 \] \[ \theta = \cos^{-1}(0.6165) \approx 0.911 \text{ radians} \] Rounded to 3 significant figures: \(\boxed{0.911}\) radians. 6. **Find vector \(\mathbf{AB} = \mathbf{OB} - \mathbf{OA}\):** \[ \mathbf{AB} = (-5 - 1)\mathbf{i} + (5 - 7)\mathbf{j} + (6 - 2)\mathbf{k} = -6 \mathbf{i} - 2 \mathbf{j} + 4 \mathbf{k} \] 7. **Given \(\mathbf{AB} = 2 \mathbf{BC}\), express \(\mathbf{BC}\):** \[ \mathbf{BC} = \frac{1}{2} \mathbf{AB} = \frac{1}{2}(-6 \mathbf{i} -2 \mathbf{j} + 4 \mathbf{k}) = -3 \mathbf{i} - \mathbf{j} + 2 \mathbf{k} \] 8. **Find position vector of C, \(\mathbf{OC}\):** Since \(\mathbf{BC} = \mathbf{OC} - \mathbf{OB}\), \[ \mathbf{OC} = \mathbf{OB} + \mathbf{BC} = (-5 \mathbf{i} + 5 \mathbf{j} + 6 \mathbf{k}) + (-3 \mathbf{i} - \mathbf{j} + 2 \mathbf{k}) = (-8 \mathbf{i} + 4 \mathbf{j} + 8 \mathbf{k}) \] 9. **Calculate magnitude of \(\mathbf{OC}\):** \[ |\mathbf{OC}| = \sqrt{(-8)^2 + 4^2 + 8^2} = \sqrt{64 + 16 + 64} = \sqrt{144} = 12 \] 10. **Calculate unit vector in direction of \(\mathbf{OC}\):** \[ \hat{u} = \frac{\mathbf{OC}}{|\mathbf{OC}|} = \frac{-8 \mathbf{i} + 4 \mathbf{j} + 8 \mathbf{k}}{12} = -\frac{2}{3} \mathbf{i} + \frac{1}{3} \mathbf{j} + \frac{2}{3} \mathbf{k} \] **Final answers:** - (i) Angle \(\angle AOB = 0.911\) radians - (ii) Unit vector in direction of \(\mathbf{OC} = -\frac{2}{3} \mathbf{i} + \frac{1}{3} \mathbf{j} + \frac{2}{3} \mathbf{k}\)