https://file.aiquickdraw.com/mathbot/file/1727283220597tpi1z1r0.jpg

Let's break down the questions visible in the image:

### Question 4
This question is asking whether a unique linear combination of **two vectors** can replace the three-vector combination discussed in a previous question. Specifically, it suggests leaving the combination of the first vector, \( a\vec{v} \), unchanged and finding another vector \( \vec{w} \), such that when added to \( \vec{v} \), it would yield the desired vector (which seems to be given by coordinates \( [7.9, 8.7] \)).

The problem asks for a comparison between linear combinations involving **three vectors** and those involving **two vectors**. To understand the situation:
- **Three Vectors**: In \( \mathbb{R}^2 \), it's typical that two vectors are sufficient to express any point in the plane, as long as they are **linearly independent**. Adding a third vector would be **redundant** (i.e., the third vector could be written as a combination of the first two).
- **Two Vectors**: The question explores whether two vectors can still form the desired vector, and asks you to find a specific second vector \( \vec{w} \) so that \( \vec{v} \) and \( \vec{w} \) together still achieve the vector \( [7.9, 8.7] \).

### Question 5
The question asks how many vectors are needed in \( \mathbb{R}^2 \) to reach any vector using linear combinations. The key points:
- In \( \mathbb{R}^2 \), you only need **two linearly independent vectors** to form any vector in the plane.
- These two vectors must be **linearly independent**, meaning that one cannot be a scalar multiple of the other.
  
The condition for linear combinations is that any vector in \( \mathbb{R}^2 \) can be obtained by appropriately scaling and adding these two vectors. Therefore, the **minimum number of vectors** needed to reach any vector in \( \mathbb{R}^2 \) is two, provided they are linearly independent.

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Would you like a detailed explanation of how to approach the solution for question 4, or do you have other questions related to linear combinations and vector spaces?

### Additional Related Questions:
1. What does it mean for two vectors to be linearly independent?
2. How do you calculate the linear combination of two vectors to get a specific result?
3. Can more than two vectors in \( \mathbb{R}^2 \) be linearly independent?
4. How do you determine if a set of vectors spans \( \mathbb{R}^2 \)?
5. Can you give an example of linearly dependent and independent vectors in \( \mathbb{R}^2 \)?

### Tip:
Remember, in any vector space, the number of vectors needed to span the space is equal to the dimension of the space.

Compare and contrast the uniqueness of the linear combination of three vectors to that of two vectors. Also, explore whether two vectors can replace three in providing a unique linear combination. Discuss how many vectors are needed to express any vector in R^2 as a linear combination.

Explore the uniqueness of linear combinations in vector spaces, specifically comparing the cases of two and three vectors. Learn how two linearly independent vectors can be sufficient to express any vector in R^2, following key principles like linear independence and the span theorem. Suitable for students in Grades 11-12.

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