1. **Problem Statement:** Given points P = (3, -1), Q = (-4, -6), R = (1, 4), and S = (2, 5), find the following vectors in component form: (i) Vector $\overrightarrow{PQ}$ (ii) $3\overrightarrow{PQ} - \overrightarrow{RS}$ (iii) $2\overrightarrow{PR} + 3\overrightarrow{PS}$ (iv) Verify if $\frac{1}{2} \overrightarrow{PQ} + \frac{5}{2} \overrightarrow{PR} = \frac{3}{2} \overrightarrow{QS}$ (v) $3^2 \overrightarrow{PS} - 4^2 \overrightarrow{SP} + \overrightarrow{QP}$ --- 2. **Problem Statement:** (i) Show that points A(1, 0), B(6, 0), and C(0, 0) are collinear. (ii) Given position vectors $\mathbf{a} = (2, -7)$ and $\mathbf{b} = (\frac{m}{2}, 11)$, find $m$ such that $\mathbf{a}$ and $\mathbf{b}$ are collinear. --- 3. **Problem Statement:** Given vectors $\mathbf{\bar{u}} = \langle -1, 1 \rangle$, $\mathbf{\bar{v}} = \langle 0, 1 \rangle$, and $\mathbf{\bar{w}} = \langle 3, 4 \rangle$: (i) Find $x$ satisfying $\mathbf{\bar{u}} - 2x = x - \mathbf{\bar{w}} + 3\mathbf{\bar{v}}$ (ii) Find $\mathbf{\bar{u}}$ and $\mathbf{\bar{v}}$ if $\mathbf{\bar{u}} + \mathbf{\bar{v}} = \langle 2, -3 \rangle$ and $3\mathbf{\bar{u}} + 2\mathbf{\bar{v}} = \langle -1, 2 \rangle$ (iii) Find the initial point of $\mathbf{v} = \langle -3, 1, 2 \rangle$ if its terminal point is $(5, 0, 1)$. --- 4. **Problem Statement:** (i) Find $m$ such that $\mathbf{a} = 3\mathbf{i} + 4\mathbf{j} - 9\mathbf{k}$ is parallel to $\mathbf{b} = \mathbf{i} + m\mathbf{j} - 3\mathbf{k}$. (ii) Find $\lambda$ such that points $P, Q, R$ with position vectors $\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$, $-2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$, and $2\lambda - \mathbf{k}$ respectively are collinear. --- 5. **Problem Statement:** (i) Given $\mathbf{a} = \mathbf{i} - 2\mathbf{j} + \mathbf{k}$, $\mathbf{b} = \mathbf{i} - \mathbf{j} - \mathbf{k}$, and $\mathbf{c} = 2\mathbf{i} + \mathbf{k}$, find a unit vector in the direction of $2\mathbf{a} - 3\mathbf{b} + \mathbf{c}$. (ii) Use vectors to find the lengths of diagonals of a parallelogram with adjacent sides $\mathbf{i} + \mathbf{j}$ and $\mathbf{i} - 2\mathbf{j}$. --- 6. **Problem Statement:** (i) Show that points with position vectors $\mathbf{i} - \mathbf{j}$, $4\mathbf{i} + 3\mathbf{j} + \mathbf{k}$, and $2\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}$ form a right-angled triangle. (ii) Show that points with position vectors $2\mathbf{i} + 3\mathbf{j} + \sqrt{3}\mathbf{k}$, $\sqrt{10}\mathbf{i} - \mathbf{j} + \sqrt{5}\mathbf{k}$, and $-3\mathbf{i} + \sqrt{3}\mathbf{j} + 2\mathbf{k}$ form an equilateral triangle. --- 7. **Problem Statement:** (i) Find $\lambda$ such that $|\mathbf{a}| = |3\mathbf{b}|$ where $\mathbf{a} = \mathbf{i} - 3\mathbf{j} + \lambda \mathbf{k}$ and $\mathbf{b} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}$. (ii) Given $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}$ and $\mathbf{b} = -3\mathbf{i} + 2\mathbf{j}$, check if $|\mathbf{a}| = |\mathbf{b}|$ or $\mathbf{a} = \mathbf{b}$. --- 8. **Problem Statement:** Given $\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}$ and $\mathbf{b} = \mathbf{i} - 3\mathbf{j} + 5\mathbf{k}$: (i) Find a vector of magnitude 5 in the direction of $\mathbf{a} - 2\mathbf{b}$. (ii) Find a vector of magnitude 3 opposite in direction to $3\mathbf{a} + \mathbf{b}$. --- 9. **Problem Statement:** The position vectors of points A and B are $\mathbf{i} - 2\mathbf{j} + \mathbf{k}$ and $2\mathbf{i} + 3\mathbf{j} - \mathbf{k}$ respectively. (i) Find the position vector of point P dividing the line segment joining A and B internally in the ratio 2:3. --- **Note:** The above restates all problems from Exercise 3.1 as requested.