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The problem asks to find vectors \( OQ \), \( OR \), \( AR \), and \( RP \) in terms of the position vectors \( a \) and \( b \) of points \( A \) and \( B \), respectively. Let’s break this down step by step:

### Given Information:

1. The position vectors of points \( A \) and \( B \) with respect to the origin \( O \) are \( \mathbf{a} \) and \( \mathbf{b} \), respectively.
2. \( 4OP = OB \) which suggests a certain proportional relationship between \( O \), \( P \), and \( B \).
3. Point \( P \) is the midpoint of \( AB \). Hence, the position vector of \( P \) is \( \frac{\mathbf{a} + \mathbf{b}}{2} \).
4. Point \( R \) lies on \( OO \) extended such that \( OR : QR = 5:3 \). This implies \( R \) divides the segment \( OQ \) in the ratio \( 5:3 \).

### Step-by-Step Solution:

#### 1. Find \( OP \) (Position vector of \( P \)):
Since \( P \) is the midpoint of \( AB \), its position vector is the average of \( \mathbf{a} \) and \( \mathbf{b} \).

\[
OP = \frac{\mathbf{a} + \mathbf{b}}{2}
\]

#### 2. Find \( OQ \) (Position vector of \( Q \)):
We are told that \( OR:QR = 5:3 \). Using section formula, we can write the position vector of \( OQ \).

Since \( OR:QR = 5:3 \), we have:

\[
OQ = \frac{5\mathbf{a} + 3\mathbf{b}}{8}
\]

#### 3. Find \( OR \):
Since \( R \) lies on the line extending from \( O \) and \( OR:QR = 5:3 \), the position vector of \( R \) can be written as a fraction of \( OQ \).

Since \( OR:QR = 5:3 \), we can write:

\[
OR = \frac{5}{8} OQ = \frac{5}{8} \times \left( \frac{5\mathbf{a} + 3\mathbf{b}}{8} \right)
\]

After expanding:

\[
OR = \frac{25\mathbf{a} + 15 \mathbf{b}}{64}
\]

#### 4. Find \( AR \):
To find \( AR \), we take the difference between \( OR \) and \( OA \) (which is just \( \mathbf{a} \)).

\[
AR = OR - OA = \left( \frac{25\mathbf{a} + 15 \mathbf{b}}{64} \right) - \mathbf{a}
\]

Simplifying:

\[
AR = \left( \frac{25\mathbf{a} + 15 \mathbf{b}}{64} \right) - \frac{64}{64}\mathbf{a}
\]
\[
AR = \frac{25\mathbf{a} + 15 \mathbf{b} - 64\mathbf{a}}{64}
\]
\[
AR = \frac{-39\mathbf{a} + 15 \mathbf{b}}{64}
\]

#### 5. Find \( RP \):
We already know \( OR \) and \( OP \), so we can find \( RP \) by subtracting \( OR \) from \( OP \):

\[
RP = OP - OR
\]
\[
RP = \frac{\mathbf{a} + \mathbf{b}}{2} - \frac{25\mathbf{a} + 15\mathbf{b}}{64}
\]

Simplifying:

\[
RP = \frac{32(\mathbf{a} + \mathbf{b}) - (25\mathbf{a} + 15 \mathbf{b})}{64}
\]
\[
RP = \frac{(32\mathbf{a} + 32\mathbf{b}) - (25\mathbf{a} + 15\mathbf{b})}{64}
\]
\[
RP = \frac{(32-25)\mathbf{a} + (32-15)\mathbf{b}}{64}
\]
\[
RP = \frac{7\mathbf{a} + 17\mathbf{b}}{64}
\]

### Final Answers:

1. \( OQ = \frac{5\mathbf{a} + 3\mathbf{b}}{8} \)
2. \( OR = \frac{25\mathbf{a} + 15 \mathbf{b}}{64} \)
3. \( AR = \frac{-39\mathbf{a} + 15 \mathbf{b}}{64} \)
4. \( RP = \frac{7\mathbf{a} + 17\mathbf{b}}{64} \)

Let me know if you would like more details or have any questions!

#### Related Questions:
1. How do you calculate the midpoint of a line segment between two points?
2. What is the section formula and how is it used in vector geometry?
3. Can you explain the concept of dividing a line segment in a given ratio in 3D space?
4. How would these results change if point \( P \) wasn’t the midpoint of \( AB \)?
5. What other geometric properties of vectors are useful in analyzing points in 3D space?

#### Tip:
Always visualize the geometry when dealing with vectors, especially when the problem involves ratios or midpoints. This can help clarify relationships between different points and segments.

The position vectors of points A and B with respect to the origin are a and b respectively. 4OP = OB. Point P is the midpoint of AB, and point R lies on OO extended so that OR:QR = 5:3. Find OQ, OR, AR, and RP in terms of a and b.

Learn how to find vectors OQ, OR, AR, and RP in terms of the position vectors a and b using midpoint and section formulas. This problem involves vector geometry concepts such as midpoints, ratios in line segments, and their applications. Suitable for Grades 10-12.

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