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Plz solve this vectors question. Maths igcse

The image appears to contain a question related to vectors involving a triangle, but it's somewhat rotated. I'll rotate and analyze the image to provide the correct solution.

### Analyzing the Problem:

#### Given Data:
- **Triangle PQR** is provided with vectors \(\overrightarrow{PQ} = \mathbf{a}\) and \(\overrightarrow{PR} = \mathbf{b}\).
- A point **S** lies on **QR** such that \(QS : SR = 1 : 2\).
- You're required to:
  1. Express \(\overrightarrow{QS}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
  2. Find \(\overrightarrow{PS}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

### Step-by-Step Solution:

#### 1. Expressing \(\overrightarrow{QS}\):
Given \(QS : SR = 1 : 2\), \(S\) divides \(QR\) in the ratio \(1:2\).

Let \(\overrightarrow{QR} = \mathbf{c}\).

Now, \(\overrightarrow{QS}\) can be found using the section formula:
\[
\overrightarrow{QS} = \frac{2\overrightarrow{Q} + 1\overrightarrow{R}}{1+2} = \frac{2\overrightarrow{Q} + \overrightarrow{R}}{3}
\]

But \(\overrightarrow{QR} = \overrightarrow{PR} - \overrightarrow{PQ} = \mathbf{b} - \mathbf{a}\).

So, \(\overrightarrow{QS}\) becomes:
\[
\overrightarrow{QS} = \frac{2(\overrightarrow{PR} - \overrightarrow{PQ}) + \overrightarrow{PR}}{3}
\]

Substitute \(\mathbf{b}\) and \(\mathbf{a}\):
\[
\overrightarrow{QS} = \frac{2(\mathbf{b} - \mathbf{a}) + \mathbf{b}}{3} = \frac{2\mathbf{b} - 2\mathbf{a} + \mathbf{b}}{3} = \frac{3\mathbf{b} - 2\mathbf{a}}{3}
\]
\[
\overrightarrow{QS} = \mathbf{b} - \frac{2}{3}\mathbf{a}
\]

#### 2. Expressing \(\overrightarrow{PS}\):
\(\overrightarrow{PS}\) is given by:
\[
\overrightarrow{PS} = \overrightarrow{PQ} + \overrightarrow{QS}
\]

Substitute the expression for \(\overrightarrow{QS}\):
\[
\overrightarrow{PS} = \mathbf{a} + \left(\mathbf{b} - \frac{2}{3}\mathbf{a}\right)
\]
\[
\overrightarrow{PS} = \mathbf{b} + \left(\mathbf{a} - \frac{2}{3}\mathbf{a}\right) = \mathbf{b} + \frac{1}{3}\mathbf{a}
\]

### Final Answers:
1. \(\overrightarrow{QS} = \mathbf{b} - \frac{2}{3}\mathbf{a}\)
2. \(\overrightarrow{PS} = \frac{1}{3}\mathbf{a} + \mathbf{b

Learn how to solve a vectors question involving triangle PQR with detailed step-by-step solutions. This problem covers vectors, section formula, and geometric properties of triangles. Suitable for IGCSE (Grade 9-10) students.

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