The image contains the following information:

$\mathbf{u} = (-2, 9), \quad \mathbf{w} = (-2, -3)$

You are asked to compute:

$-2\mathbf{u} - 5\mathbf{w} = \, ?$

Step-by-Step Solution:

1. First, let's compute $-2\mathbf{u}$. To do this, multiply each component of vector $\mathbf{u} = (-2, 9)$ by $-2$:

$-2\mathbf{u} = -2 \times (-2, 9) = (4, -18)$

2. Now, compute $-5\mathbf{w}$. Multiply each component of vector $\mathbf{w} = (-2, -3)$ by $-5$:

$-5\mathbf{w} = -5 \times (-2, -3) = (10, 15)$

3. Finally, add the results of the two vectors:

$-2\mathbf{u} - 5\mathbf{w} = (4, -18) + (10, 15) = (4 + 10, -18 + 15) = (14, -3)$

Thus:

$-2\mathbf{u} - 5\mathbf{w} = (14, -3)$

Would you like further details on this? Here are 5 related questions you can explore:

1. How do you perform vector addition and scalar multiplication?
2. What are some real-world applications of vector operations?
3. How do you calculate the magnitude of a vector?
4. Can vectors in 3D space be solved using similar operations?
5. What is the difference between dot product and cross product?

Tip: When working with vectors, always ensure that the operations (like scalar multiplication and addition) are applied component-wise.