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The problem asks you to represent and calculate the magnitude (or modulus) of several vectors. The vectors given are:

1. \(\vec{u} = (-2, 2)\)
2. \(\vec{v} = (1, 0)\)
3. \(\vec{r} = \left(-1, \frac{3}{2}\right)\)
4. \(\vec{s} = (\sqrt{2}, -\sqrt{2})\)

### Steps to Calculate the Magnitude of a Vector

For any vector \(\vec{a} = (x, y)\), the magnitude (or modulus) \(|\vec{a}|\) is calculated using the formula:
\[
|\vec{a}| = \sqrt{x^2 + y^2}
\]

Let's calculate the magnitude of each vector:

1. **Vector \(\vec{u} = (-2, 2)\):**
   \[
   |\vec{u}| = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
   \]

2. **Vector \(\vec{v} = (1, 0)\):**
   \[
   |\vec{v}| = \sqrt{1^2 + 0^2} = \sqrt{1} = 1
   \]

3. **Vector \(\vec{r} = \left(-1, \frac{3}{2}\right)\):**
   \[
   |\vec{r}| = \sqrt{\left(-1\right)^2 + \left(\frac{3}{2}\right)^2} = \sqrt{1 + \frac{9}{4}} = \sqrt{\frac{4}{4} + \frac{9}{4}} = \sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2}
   \]

4. **Vector \(\vec{s} = (\sqrt{2}, -\sqrt{2})\):**
   \[
   |\vec{s}| = \sqrt{\left(\sqrt{2}\right)^2 + \left(-\sqrt{2}\right)^2} = \sqrt{2 + 2} = \sqrt{4} = 2
   \]

### Summary of Magnitudes:
- \(|\vec{u}| = 2\sqrt{2}\)
- \(|\vec{v}| = 1\)
- \(|\vec{r}| = \frac{\sqrt{13}}{2}\)
- \(|\vec{s}| = 2\)

Would you like to know how to graph these vectors or have any further questions?

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Here are five questions to deepen your understanding:

1. How do you find the direction of a vector using its components?
2. What is the geometric interpretation of the magnitude of a vector?
3. Can you think of an example where the magnitude of a vector equals zero?
4. How does the magnitude change if a vector is scaled by a constant factor?
5. What happens to the magnitude of a vector when it is rotated in a plane?

**Tip:** The magnitude of a vector always represents the distance from the origin to the point described by the vector.

Representa y calcula el módulo de los siguientes vectores: \(\vec{u} = (-2, 2)\), \(\vec{v} = (1, 0)\), \(\vec{r} = \left(-1, \frac{3}{2}\right)\), \(\vec{s} = (\sqrt{2}, -\sqrt{2})\).

This problem involves calculating the magnitude of vectors \(\vec{u} = (-2, 2)\), \(\vec{v} = (1, 0)\), \(\vec{r} = \left(-1, \frac{3}{2}\right)\), and \(\vec{s} = (\sqrt{2}, -\sqrt{2})\) using the Euclidean distance formula. This problem is suitable for students in Grades 9-12.

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