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The exercise involves determining all points on the line \( L \) that are twice as far from \( (1,3) \) as from \( (7,5) \). The problem is asking for points \( p \) and \( q \) that satisfy this condition, and the goal is to calculate vectors \( V \) and \( W \), as well as to determine the scaling factors involved.

Let's break down the key steps:

1. **Sketch the vectors \( V \) and \( W \):**
   - Vector \( V = (7,5) - p \), which represents the vector from \( p \) to \( (7,5) \).
   - Vector \( W = q - (7,5) \), which represents the vector from \( (7,5) \) to \( q \).

2. **Determine scaling factors**: 
   - The problem states that the distance between \( p \) and \( (1,3) \) is twice that between \( p \) and \( (7,5) \).
   - Likewise, you need to find a similar relationship for the vector involving \( q \).

For each part of the problem:

- **Part (a):** You should graphically represent the vectors.
- **Part (b):** The vectors \( V \) and \( W \) are described as multiples of \( (7,5) - (1,3) \), so you need to determine their relationship.
- **Part (c):** Compute the coordinates of points \( p \) and \( q \) by solving the distance ratio equations for the given conditions.

Would you like a detailed breakdown of each part with corresponding calculations?

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Here are 5 questions to deepen understanding:
1. How do we compute the distance between two points in a plane?
2. What is the relationship between vector scaling and distances?
3. How can you express the condition of "twice as far" using vector norms?
4. What kind of equation would you set up to find points equidistant from two points?
5. What geometric shape describes the locus of points equidistant from two given points?

**Tip:** When working with vectors, always remember that the magnitude of a vector represents the distance between the two points involved.

Determine all points on line L that are twice as far from (1,3) as they are from (7,5). The problem involves finding points p and q such that the distance from p to (1,3) is twice the distance from p to (7,5), and similarly for q.

This problem explores vector analysis to find points p and q on a line where the distance between p and (1,3) is twice the distance between p and (7,5). The solution involves using the distance formula, vector representation, and proportional relationships, making it ideal for students in Grades 10-12.

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