https://r2.erweima.ai/mathbot/image/1725472484483iwdv2jkj.jpg

The problem asks to represent and calculate the magnitudes (or moduli) of the following vectors:

\[
\vec{v} = (-2, 2), \quad \vec{w} = (1, 0), \quad \vec{r} = \left(-1, \frac{2}{3}\right), \quad \vec{s} = \left(\sqrt{2}, -\sqrt{2}\right)
\]

### Step 1: Understanding the magnitude (modulus) of a vector
The magnitude \( |\vec{a}| \) of a vector \( \vec{a} = (x, y) \) in a 2D plane is given by the formula:
\[
|\vec{a}| = \sqrt{x^2 + y^2}
\]

### Step 2: Calculate the magnitude for each vector

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

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

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

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

### Summary of Magnitudes:
- \( |\vec{v}| = 2\sqrt{2} \)
- \( |\vec{w}| = 1 \)
- \( |\vec{r}| = \frac{\sqrt{13}}{3} \)
- \( |\vec{s}| = 2 \)

Would you like to discuss the graphical representation of these vectors or have any other questions? 

Here are 5 related questions for further exploration:

1. How do you find the direction (angle) of a vector given its components?
2. What would the magnitudes of these vectors be if the components were in three dimensions?
3. How can you determine if two vectors are perpendicular using their components?
4. What does the magnitude of a vector represent in physical applications?
5. How do you add two vectors geometrically and algebraically?

**Tip:** Always ensure the vector components are squared individually before summing them when calculating the magnitude. This helps avoid mistakes, especially when dealing with negative components or fractions.

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

This guide explains how to calculate the magnitude of 2D vectors like (-2, 2), (1, 0), (-1, 2/3), and (√2, -√2). Learn the steps to solve vector magnitude problems using the Pythagorean theorem. Suitable for students in Grades 9-12.

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