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.