The problem asks us to find the projection of vector $\textbf{w}$ onto vector $\textbf{u}$, given $\textbf{u} = \textbf{i} + 2\textbf{j}$ and $\textbf{w} = \textbf{i} + 5\textbf{j}$. We need to compute $proj_{\textbf{u}}\textbf{w}$.

1. Problem Description

The problem asks us to find the projection of vector $\textbf{w}$ onto vector $\textbf{u}$, given $\textbf{u} = \textbf{i} + 2\textbf{j}$ and $\textbf{w} = \textbf{i} + 5\textbf{j}$. We need to compute $proj_{\textbf{u}}\textbf{w}$.

2. Solution Steps

First, we recall the formula for the projection of a vector $\textbf{w}$ onto a vector $\textbf{u}$:

$$proj_{\textbf{u}}\textbf{w} = \frac{\textbf{w} \cdot \textbf{u}}{\|\textbf{u}\|^2} \textbf{u}$$

We are given $\textbf{u} = \textbf{i} + 2\textbf{j} = <1, 2>$ and $\textbf{w} = \textbf{i} + 5\textbf{j} = <1, 5>$.

We need to find the dot product $\textbf{w} \cdot \textbf{u}$:

$$\textbf{w} \cdot \textbf{u} = (1)(1) + (5)(2) = 1 + 10 = 11$$

Next, we need to find the squared magnitude of $\textbf{u}$, which is $\|\textbf{u}\|^2$:

$$\|\textbf{u}\|^2 = (1)^2 + (2)^2 = 1 + 4 = 5$$

Now, we plug these values into the projection formula:

$$proj_{\textbf{u}}\textbf{w} = \frac{11}{5} \textbf{u} = \frac{11}{5} <1, 2> = <\frac{11}{5}, \frac{22}{5}>$$

So, $proj_{\textbf{u}}\textbf{w} = \frac{11}{5}\textbf{i} + \frac{22}{5}\textbf{j}$.

3. Final Answer

$\frac{11}{5}\textbf{i} + \frac{22}{5}\textbf{j}$