https://file.aiquickdraw.com/mathbot/file/17278615533416qsk3znx.jpg

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.

Compute 5u + 2v - 2w where u = [3, 4, 5], v = [4, 8, 1], w = [5, 5, 9].

Learn how to compute the vector expression 5u + 2v - 2w where u = [3, 4, 5], v = [4, 8, 1], and w = [5, 5, 9]. This solution covers scalar multiplication and vector addition with a step-by-step guide.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation