1. **State the problem:** We have triangle OAB with points P and Q defined as follows: - P is the midpoint of OA. - Q lies on OB such that \(\overrightarrow{OQ} = n\mathbf{b}\). - Given vectors \(\overrightarrow{OA} = 12\mathbf{a}\) and \(\overrightarrow{OB} = 8\mathbf{b}\). - BR and PQR are straight lines. We need to: a) Express \(\overrightarrow{AB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). b) Find the value of \(n\) given \(AB : BR = 1 : 2\). 2. **Express \(\overrightarrow{AB}\):** \[\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = 8\mathbf{b} - 12\mathbf{a}\] 3. **Find coordinates of points:** - Since P is midpoint of OA: \[\overrightarrow{OP} = \frac{1}{2} \overrightarrow{OA} = \frac{1}{2} \times 12\mathbf{a} = 6\mathbf{a}\] - Q lies on OB, so: \[\overrightarrow{OQ} = n \times 8\mathbf{b} = 8n \mathbf{b}\] 4. **Express vectors for points B and R:** - B lies on AR, but we need to find \(\overrightarrow{OB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\). Given \(\overrightarrow{OB} = 8\mathbf{b}\). - R lies on line BR such that \(AB : BR = 1 : 2\). 5. **Find \(\overrightarrow{BR}\):** Since \(AB : BR = 1 : 2\), point R divides line segment AB extended beyond B in ratio 1:2. Using section formula for point R dividing AB externally in ratio 1:2: \[ \overrightarrow{OR} = \frac{2 \overrightarrow{OA} - 1 \overrightarrow{OB}}{2 - 1} = 2 \overrightarrow{OA} - \overrightarrow{OB} = 2 \times 12\mathbf{a} - 8\mathbf{b} = 24\mathbf{a} - 8\mathbf{b} \] 6. **Since P, Q, R are collinear, vectors \(\overrightarrow{PQ}\) and \(\overrightarrow{QR}\) are parallel:** \[ \overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} = 8n \mathbf{b} - 6 \mathbf{a} \] \[ \overrightarrow{QR} = \overrightarrow{OR} - \overrightarrow{OQ} = (24 \mathbf{a} - 8 \mathbf{b}) - 8n \mathbf{b} = 24 \mathbf{a} - 8(1 + n) \mathbf{b} \] 7. **Set \(\overrightarrow{PQ}\) and \(\overrightarrow{QR}\) parallel:** There exists a scalar \(k\) such that: \[ \overrightarrow{PQ} = k \overrightarrow{QR} \] Equate components: - For \(\mathbf{a}\): \[-6 = 24k \implies k = -\frac{1}{4}\] - For \(\mathbf{b}\): \[8n = k \times -8(1 + n) = -\frac{1}{4} \times -8(1 + n) = 2(1 + n)\] 8. **Solve for \(n\):** \[ 8n = 2(1 + n) \\ 8n = 2 + 2n \\ 8n - 2n = 2 \\ 6n = 2 \\ \Rightarrow n = \frac{1}{3} \] **Final answers:** - \(\overrightarrow{AB} = 8\mathbf{b} - 12\mathbf{a}\) - \(n = \frac{1}{3}\)