Given 𝑎1=−3 and 𝑎2=4, what vector is formed by the linear combination 𝑎1v1+𝑎2v2?

𝑎1v1+𝑎2v2=[]

v2=[−3−7]

v1=[8 -7]

The vector formed by the linear combination \( a_1 v_1 + a_2 v_2 \) can be calculated by performing the scalar multiplication of each vector and then adding the results together.

Given:

- \( a_1 = -3 \), \( a_2 = 4 \)
- \( v_1 = \begin{bmatrix} 8 \\ -7 \end{bmatrix} \), \( v_2 = \begin{bmatrix} -3 \\ -7 \end{bmatrix} \)

### Step 1: Multiply each vector by its corresponding scalar

\[
a_1 v_1 = -3 \times \begin{bmatrix} 8 \\ -7 \end{bmatrix} = \begin{bmatrix} -24 \\ 21 \end{bmatrix}
\]

\[
a_2 v_2 = 4 \times \begin{bmatrix} -3 \\ -7 \end{bmatrix} = \begin{bmatrix} -12 \\ -28 \end{bmatrix}
\]

### Step 2: Add the resulting vectors

\[
a_1 v_1 + a_2 v_2 = \begin{bmatrix} -24 \\ 21 \end{bmatrix} + \begin{bmatrix} -12 \\ -28 \end{bmatrix} = \begin{bmatrix} -36 \\ -7 \end{bmatrix}
\]

### Final Answer:

\[
a_1 v_1 + a_2 v_2 = \begin{bmatrix} -36 \\ -7 \end{bmatrix}
\]

Let me know if you want more details or have any questions.

Here are five related questions for further practice:
1. What is the magnitude of the vector \( \begin{bmatrix} -36 \\ -7 \end{bmatrix} \)?
2. What is the dot product of \( v_1 \) and \( v_2 \)?
3. If \( v_3 = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \), what is \( a_1 v_1 + a_2 v_2 + v_3 \)?
4. What is the angle between \( v_1 \) and \( v_2 \)?
5. What happens if \( a_1 = a_2 \) in the linear combination?

**Tip:** When adding vectors, always add their corresponding components.

Given 𝑎1 = −3 and 𝑎2 = 4, what vector is formed by the linear combination 𝑎1v1 + 𝑎2v2? v1 = [8, -7], v2 = [-3, -7]

This problem explores how to compute the linear combination of two vectors using given scalar values. Learn to apply scalar multiplication and vector addition to find the resulting vector, v = [-36, -7], suitable for high school students studying linear algebra.

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