We are tasked with computing $5\mathbf{u} + 2\mathbf{v} - 2\mathbf{w}$, where:

$\mathbf{u} = \begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 4 \\ 8 \\ 1 \end{bmatrix}, \quad \mathbf{w} = \begin{bmatrix} 5 \\ 5 \\ 9 \end{bmatrix}$

Step 1: Multiply each vector by its scalar coefficient.

• $5\mathbf{u} = 5 \times \begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix} = \begin{bmatrix} 15 \\ 20 \\ 25 \end{bmatrix}$
• $2\mathbf{v} = 2 \times \begin{bmatrix} 4 \\ 8 \\ 1 \end{bmatrix} = \begin{bmatrix} 8 \\ 16 \\ 2 \end{bmatrix}$
• $-2\mathbf{w} = -2 \times \begin{bmatrix} 5 \\ 5 \\ 9 \end{bmatrix} = \begin{bmatrix} -10 \\ -10 \\ -18 \end{bmatrix}$

Step 2: Add the vectors.

Now, sum the results:

$5\mathbf{u} + 2\mathbf{v} - 2\mathbf{w} = \begin{bmatrix} 15 \\ 20 \\ 25 \end{bmatrix} + \begin{bmatrix} 8 \\ 16 \\ 2 \end{bmatrix} + \begin{bmatrix} -10 \\ -10 \\ -18 \end{bmatrix}$

Perform the addition element-wise:

$= \begin{bmatrix} 15 + 8 - 10 \\ 20 + 16 - 10 \\ 25 + 2 - 18 \end{bmatrix} = \begin{bmatrix} 13 \\ 26 \\ 9 \end{bmatrix}$

Final Answer:

$\begin{bmatrix} 13 \\ 26 \\ 9 \end{bmatrix}$

Would you like further details on the steps? Here are some related questions for practice:

1. How would the result change if we used $3\mathbf{u} + 4\mathbf{v} - \mathbf{w}$?
2. What happens if we swap the roles of vectors $\mathbf{u}$ and $\mathbf{w}$ in the original expression?
3. How do you multiply a scalar by a vector?
4. What is the geometric interpretation of vector addition?
5. How could we check the accuracy of our result?

Tip: When performing matrix or vector arithmetic, always handle each component separately and ensure the dimensions match for proper addition.