https://file.aiquickdraw.com/mathbot/file/1726010365010xfczxmwh.jpg

I can see the image you uploaded, which contains a linear combination problem. Let's solve it step by step.

Given:
\[
\mathbf{u} = \begin{pmatrix} 2 \\ 4 \\ 2 \end{pmatrix}, 
\mathbf{v} = \begin{pmatrix} -2 \\ 1 \\ 5 \end{pmatrix}, 
\mathbf{w} = \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix}
\]

You are asked to compute:
\[
2\mathbf{u} + 3\mathbf{v} + 2\mathbf{w}
\]

First, let's compute each term:

1. \( 2\mathbf{u} = 2 \times \begin{pmatrix} 2 \\ 4 \\ 2 \end{pmatrix} = \begin{pmatrix} 4 \\ 8 \\ 4 \end{pmatrix} \)
2. \( 3\mathbf{v} = 3 \times \begin{pmatrix} -2 \\ 1 \\ 5 \end{pmatrix} = \begin{pmatrix} -6 \\ 3 \\ 15 \end{pmatrix} \)
3. \( 2\mathbf{w} = 2 \times \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 10 \\ 2 \\ 0 \end{pmatrix} \)

Now, summing these vectors:
\[
\begin{pmatrix} 4 \\ 8 \\ 4 \end{pmatrix} + \begin{pmatrix} -6 \\ 3 \\ 15 \end{pmatrix} + \begin{pmatrix} 10 \\ 2 \\ 0 \end{pmatrix}
= \begin{pmatrix} (4 - 6 + 10) \\ (8 + 3 + 2) \\ (4 + 15 + 0) \end{pmatrix}
= \begin{pmatrix} 8 \\ 13 \\ 19 \end{pmatrix}
\]

Thus, the result of the linear combination is:
\[
\mathbf{r} = \begin{pmatrix} 8 \\ 13 \\ 19 \end{pmatrix}
\]

Would you like further details on this process or have any questions?

Here are 5 related questions to expand this topic:
1. What is a general definition of a linear combination in vector spaces?
2. How can linear independence of vectors be determined?
3. Can you compute a different linear combination using scalar multiples?
4. What role do linear combinations play in solving systems of linear equations?
5. How does this relate to the span of vectors?

**Tip**: When solving vector operations, always ensure each term is correctly multiplied before performing any addition or subtraction.

If u = (2, 4, 2), v = (-2, 1, 5), and w = (5, 1, 0), calculate the following linear combination: 2u + 3v + 2w.

Learn how to calculate the linear combination of vectors u = (2, 4, 2), v = (-2, 1, 5), and w = (5, 1, 0) to find 2u + 3v + 2w. Step-by-step solution with vector addition and scalar multiplication, suitable for students in Grades 9-12.

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