https://file.aiquickdraw.com/mathbot/file/1727448687539g647ndyv.jpg

The problem you've shared involves vector geometry. Let's break it down step by step:

### Given:
- Position vectors of points \( A \) and \( B \) are given by vectors \( \mathbf{a} \) and \( \mathbf{b} \), respectively.
- Point \( P \) is the midpoint of line segment \( AB \), so:
  \[
  \mathbf{P} = \frac{\mathbf{a} + \mathbf{b}}{2}
  \]
- \( OR:QR = 5:3 \), where point \( R \) lies on the line \( OO \), meaning point \( O \) is the origin.

We are tasked with finding \( \mathbf{OQ}, \mathbf{OR}, \mathbf{AR}, \) and \( \mathbf{RP} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).

### Step 1: Understanding Point \( P \)'s Position
Point \( P \) is the midpoint of \( AB \), so the position vector of \( P \) is:
\[
\mathbf{P} = \frac{\mathbf{a} + \mathbf{b}}{2}
\]

### Step 2: Ratio of OR to QR
We are told that \( OR : QR = 5:3 \). This implies that the line segment \( OQ \) is divided in the ratio \( 5:3 \), where \( R \) lies between \( O \) and \( Q \).

Using the section formula for internal division, if \( OR:QR = 5:3 \), the position vector of \( Q \) is:
\[
\mathbf{Q} = \frac{5\mathbf{R} + 3\mathbf{O}}{5+3} = \frac{5\mathbf{R}}{8}
\]
(because the origin is at \( O = 0 \)).

### Step 3: Find OQ, OR, AR, and RP
Let me solve this step-by-step for each expression.

I will compute these values, including the detailed solution.Here are the expressions for each of the required vectors:

1. **Position of \( \mathbf{OQ} \):**
   \[
   \mathbf{OQ} = \frac{8}{5} \mathbf{OR}
   \]
   This comes from the ratio \( OR:QR = 5:3 \), so \( OQ \) is \( \frac{8}{5} \times OR \).

2. **Position of \( \mathbf{OR} \):**
   \[
   \mathbf{OR} = \mathbf{OR}
   \]
   This is simply the position vector of point \( R \) from the origin.

3. **Position of \( \mathbf{AR} \):**
   \[
   \mathbf{AR} = \mathbf{OR} - \mathbf{a}
   \]
   This represents the vector from \( A \) to \( R \).

4. **Position of \( \mathbf{RP} \):**
   \[
   \mathbf{RP} = \frac{\mathbf{a} + \mathbf{b}}{2} - \mathbf{OR}
   \]
   The vector from \( R \) to the midpoint \( P \) of \( AB \).

Would you like any further clarifications or detailed steps on any part of this?

### Related Questions:
1. How do we derive the section formula for vectors?
2. What are the properties of midpoints in vector geometry?
3. How does the ratio \( OR:QR \) affect the total vector distance?
4. How can we apply this method to different geometric shapes?
5. What are some real-world applications of vector geometry?

### Tip:
Always double-check the ratio-based calculations, as small errors in interpreting ratios can lead to incorrect vector expressions.

The position vectors of points A and B with respect to the origin are a and b respectively. 4OP = O8. 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.

Solve a vector geometry problem where the position vectors of points A and B are given, with the midpoint P and a ratio OR:QR = 5:3. Find OQ, OR, AR, and RP in terms of a and b using the midpoint formula and section formula. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation