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.