Question Six: Given that a=3i+5j, b=7i+3j, and c=-3i+2j, and OC=7i+5j. The position vectors of three students are described as OA=6i+2j, OB=3i+12j, and OC=7i+5j. Evaluate the following: AC. ii) |BC|. iii) AB + 2AC.

Huu hapa ni utatuzi wa swali la Sita:

Swali la Sita

a) Kutokana na vekta $\mathbf{a}=3\mathbf{i}+5\mathbf{j}$, $\mathbf{b}=7\mathbf{i}+3\mathbf{j}$, na $\mathbf{c}=-3\mathbf{i}+2\mathbf{j}$, tafuta $\mathbf{r}= \mathbf{a}+\mathbf{b}-\mathbf{c}$, ukubwa wake $|\mathbf{r}|$ na mwelekeo wake.

Step 1: Tafuta vekta $\mathbf{r}$.

$$\mathbf{r} = \mathbf{a} + \mathbf{b} - \mathbf{c}$$ $$\mathbf{r} = (3\mathbf{i} + 5\mathbf{j}) + (7\mathbf{i} + 3\mathbf{j}) - (-3\mathbf{i} + 2\mathbf{j})$$ $$\mathbf{r} = (3+7-(-3))\mathbf{i} + (5+3-2)\mathbf{j}$$ $$\mathbf{r} = (3+7+3)\mathbf{i} + (8-2)\mathbf{j}$$ $$\mathbf{r} = 13\mathbf{i} + 6\mathbf{j}$$

Step 2: Tafuta ukubwa (magnitude) wa $\mathbf{r}$, $|\mathbf{r}|$.

$$|\mathbf{r}| = \sqrt{13^2 + 6^2}$$ $$|\mathbf{r}| = \sqrt{169 + 36}$$ $$|\mathbf{r}| = \sqrt{205}$$

Step 3: Tafuta mwelekeo (direction) wa $\mathbf{r}$.
Mwelekeo hutolewa na pembe $\theta$ na mhimili chanya wa x.

$$\tan \theta = \frac{6}{13}$$ $$\theta = \arctan\left(\frac{6}{13}\right)$$ $$\theta \approx 24.77^\circ$$ $$\boxed{\mathbf{r} = 13\mathbf{i} + 6\mathbf{j}, \quad |\mathbf{r}| = \sqrt{205}, \quad \text{mwelekeo} \approx 24.77^\circ}$$