https://file.aiquickdraw.com/mathbot/file/1727277387765w4i2wxa5.jpg

Let's go through the problem step-by-step based on the image you provided:

### Part a
In this question, you are tasked with creating a **linear combination** of two vectors, \(\vec{v}\) and \(\vec{w}\). 
You start with:

\[
\vec{v} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}, \quad \vec{w} = \begin{bmatrix} 3 \\ 8 \end{bmatrix}
\]

You need to find scalars \(a\) and \(b\) such that:

\[
a\vec{v} + b\vec{w} = \vec{target}
\]

Where the target vector appears to be \(\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}\).

- **Solution:**
By expanding the equation, you get:

\[
a \begin{bmatrix} 2 \\ 1 \end{bmatrix} + b \begin{bmatrix} 3 \\ 8 \end{bmatrix} = \begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}
\]

This gives two equations:

1. \(2a + 3b = 9.4\)
2. \(a + 8b = 6.4\)

You can now solve these simultaneous equations to find \(a\) and \(b\).

**Steps**:
1. Solve for \(a\) in terms of \(b\) using the second equation: 
   
   \( a = 6.4 - 8b \)

2. Substitute this into the first equation:
   
   \( 2(6.4 - 8b) + 3b = 9.4 \)

3. Solve for \(b\) and then back-calculate \(a\).

---

### Part b
This part asks you to place your target vector \(\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}\) and then modify \(\vec{w}\) so that it is not possible to reach the target with any linear combination of \(\vec{v}\) and \(\vec{w}\).

- **Key Concept:**
   This question tests the concept of **linear dependence**. If \(\vec{v}\) and \(\vec{w}\) are **linearly dependent**, it means they are scalar multiples of each other. In this case, no combination of \(\vec{v}\) and \(\vec{w}\) can span the 2D plane and thus you can't reach certain vectors, like your target vector.

If the new \(\vec{w}\) is a multiple of \(\vec{v}\), then you can't use them to form the target vector \(\begin{bmatrix} 9.4 \\ 6.4 \end{bmatrix}\).

---

### Part c
The final part asks a general question about whether any two vectors in \(\mathbb{R}^2\) can be combined to produce any vector in \(\mathbb{R}^2\).

- **Answer:**
   In general, if two vectors are **linearly independent**, then you can combine them in a linear combination to get any vector in \(\mathbb{R}^2\). If the vectors are **linearly dependent**, they lie on the same line, and thus you can only reach vectors on that line.

Would you like me to help solve the equations from part (a) step-by-step?

### 5 Related Questions:
1. How do you solve systems of linear equations using substitution or elimination?
2. What does it mean for vectors to be linearly independent?
3. Can a set of vectors span a plane even if one of them is a scalar multiple of the other?
4. How do linear combinations relate to vector spaces and their dimensions?
5. What is the geometric interpretation of linear combinations of vectors?

### Tip:
Always visualize the vectors in \(\mathbb{R}^2\) to help understand linear independence and dependence better! It can clarify which areas of the plane they can span together.

Choose two new vectors to use for vectors v and w, redefining them to fit into the graphing window. Then, perform a linear combination to reach a target vector and determine if it's possible to define another vector for w such that a linear combination does not reach the target vector. Discuss the geometric relationship between v and w, and determine whether two vectors in R^2 can always combine to generate any vector in R^2.

Explore linear combinations of vectors in R^2 using two given vectors and solving systems of equations. Learn about linear dependence, geometric relationships, and whether two vectors can span the entire 2D plane. Suitable for advanced high school and college-level students.

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