Calculate the dot product and magnitudes of vectors r = 2i + 5j and s = 4i + 6j.

Step 1: Find the dot product of the vectors.
Given vectors are $\vec{r} = 2\vec{i} + 5\vec{j}$ and $\vec{s} = 4\vec{i} + 6\vec{j}$.
The dot product $\vec{r} \cdot \vec{s}$ is:

$$\vec{r} \cdot \vec{s} = (2)(4) + (5)(6) = 8 + 30 = 38$$

Step 2: Find the magnitudes of the vectors.
The magnitude of $\vec{r}$ is $|\vec{r}|$:

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