Am not understanding kiswahili

From the given data, we have the position vectors:

$$\vec{OA} = 2\mathbf{i} + 3\mathbf{j}$$ $$\vec{OB} = 10\mathbf{i} + 9\mathbf{j}$$ $$\vec{OC} = 14\mathbf{i} + 5\mathbf{j}$$

The vector for the line from substation A to village B is:

$$\vec{AB} = \vec{OB} - \vec{OA} = (10-2)\mathbf{i} + (9-3)\mathbf{j} = 8\mathbf{i} + 6\mathbf{j}$$

The length of the distribution line AB is:

$$|\vec{AB}| = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \text{ km}$$

The vector for the line from substation A to village C is:

$$\vec{AC} = \vec{OC} - \vec{OA} = (14-2)\mathbf{i} + (5-3)\mathbf{j} = 12\mathbf{i} + 2\mathbf{j}$$

The length of the distribution line AC is:

$$|\vec{AC}| = \sqrt{12^2 + 2^2} = \sqrt{144 + 4} = \sqrt{148} = 2\sqrt{37} \text{ km} \approx 12.17 \text{ km}$$

We use the dot product formula: $\vec{AB} \cdot \vec{AC} = |\vec{AB}| |\vec{AC}| \cos\phi$.

$$\vec{AB} \cdot \vec{AC} = (8)(12) + (6)(2) = 96 + 12 = 108$$ $$\cos\phi = \frac{108}{10 \cdot 2\sqrt{37}} = \frac{108}{20\sqrt{37}} = \frac{27}{5\sqrt{37}}$$ $$\phi = \arccos\left(\frac{27}{5\sqrt{37}}\right) \approx 27.4^\circ$$

The requirement is that the angle between the two distribution lines from the substation must be less than $45^\circ$. Since $27.4^\circ < 45^\circ$, this condition is met.

The engineer used the equation $4\sin^2\theta - 3 = 0$ to represent the inclination of the distribution line from A to B.

$$4\sin^2\theta = 3$$ $$\sin^2\theta = \frac{3}{4}$$ $$\sin\theta = \pm\sqrt{\frac{3}{4}} = \pm\frac{\sqrt{3}}{2}$$

Case 1: $\sin\theta = \frac{\sqrt{3}}{2}$
This gives $\theta = 60^\circ$ or $\theta = 180^\circ - 60^\circ = 120^\circ$.
Case 2: $\sin\theta = -\frac{\sqrt{3}}{2}$
This gives $\theta = 180^\circ + 60^\circ = 240^\circ$ or $\theta = 360^\circ - 60^\circ = 300^\circ$.
The possible values for $\theta$ are $60^\circ, 120^\circ, 240^\circ, 300^\circ$.
The distribution line from A to B has vector $\vec{AB} = 8\mathbf{i} + 6\mathbf{j}$, which points into the first quadrant (positive slope). Therefore, its angle of inclination should be between $0^\circ$ and $90^\circ$. Among the values found, only $60^\circ$ is in the first quadrant.

Summary of Guidance:

• The length of the distribution line from substation A to village B is $\boxed{\text{10 km}}$.
• The length of the distribution line from substation A to village C is $\boxed{2\sqrt{37} \text{ km}}$.
• The angle between the distribution lines AB and AC at substation A is approximately $\boxed{27.4^\circ}$, which satisfies the condition of being less than $45^\circ$.
• Based on the engineer's system, the appropriate angle of inclination for the distribution line from substation A to village B is $\boxed{60^\circ}$.