https://file.aiquickdraw.com/mathbot/file/1728755851057fbcz9z4r.jpg

Let's break this down step by step.

### Given:
- \(\vec{x} = \begin{bmatrix} -1 \\ -1 \end{bmatrix}\)
- \(\vec{y} = \begin{bmatrix} 1 \\ -5 \end{bmatrix}\)

We are asked to find the following vectors:
1. \(\vec{v} = 7\vec{x}\)
2. \(\vec{u} = 3\vec{y}\)
3. \(\vec{w} = 7\vec{x} + 3\vec{y}\)

### Step 1: Calculate \(\vec{v} = 7\vec{x}\)

\[
\vec{v} = 7 \begin{bmatrix} -1 \\ -1 \end{bmatrix} = \begin{bmatrix} 7 \times -1 \\ 7 \times -1 \end{bmatrix} = \begin{bmatrix} -7 \\ -7 \end{bmatrix}
\]

### Step 2: Calculate \(\vec{u} = 3\vec{y}\)

\[
\vec{u} = 3 \begin{bmatrix} 1 \\ -5 \end{bmatrix} = \begin{bmatrix} 3 \times 1 \\ 3 \times -5 \end{bmatrix} = \begin{bmatrix} 3 \\ -15 \end{bmatrix}
\]

### Step 3: Calculate \(\vec{w} = 7\vec{x} + 3\vec{y}\)

\[
\vec{w} = 7\vec{x} + 3\vec{y} = \begin{bmatrix} -7 \\ -7 \end{bmatrix} + \begin{bmatrix} 3 \\ -15 \end{bmatrix} = \begin{bmatrix} -7 + 3 \\ -7 + (-15) \end{bmatrix} = \begin{bmatrix} -4 \\ -22 \end{bmatrix}
\]

### Final Answers:
1. \(\vec{v} = \begin{bmatrix} -7 \\ -7 \end{bmatrix}\)
2. \(\vec{u} = \begin{bmatrix} 3 \\ -15 \end{bmatrix}\)
3. \(\vec{w} = \begin{bmatrix} -4 \\ -22 \end{bmatrix}\)

Let me know if you'd like more details or have any questions!

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Here are 5 related questions for further practice:
1. What is the magnitude of vector \(\vec{v}\)?
2. Find the dot product of \(\vec{x}\) and \(\vec{y}\).
3. Calculate the unit vector of \(\vec{u}\).
4. What is the result of adding vectors \(\vec{v}\) and \(\vec{u}\)?
5. How would you find the angle between \(\vec{x}\) and \(\vec{y}\)?

**Tip:** When multiplying a scalar by a vector, each component of the vector is multiplied by the scalar.

Let \( \vec{x} = \begin{bmatrix} -1 \\ -1 \end{bmatrix} \) and \( \vec{y} = \begin{bmatrix} 1 \\ -5 \end{bmatrix} \). Find the following vectors: \( \vec{v} = 7\vec{x} \), \( \vec{u} = 3\vec{y} \), and \( \vec{w} = 7\vec{x} + 3\vec{y} \).

Solve vector equations using scalar multiplication and vector addition. Given vectors \( \vec{x} = \begin{bmatrix} -1 \\ -1 \end{bmatrix} \) and \( \vec{y} = \begin{bmatrix} 1 \\ -5 \end{bmatrix} \), find \( \vec{v} = 7\vec{x} \), \( \vec{u} = 3\vec{y} \), and \( \vec{w} = 7\vec{x} + 3\vec{y} \). Suitable for high school students.

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