https://r2.erweima.ai/mathbot/image/1725422293962i7mu8f1w.jpg

The image shows the following mathematical expression:

\[
\|\vec{v}\|^2 = -1^2 + \frac{2^2}{3}
\]

Let's solve this step by step.

1. First, calculate \(-1^2\):
   \[
   -1^2 = (-1) \times (-1) = 1
   \]

2. Next, calculate \(\frac{2^2}{3}\):
   \[
   2^2 = 2 \times 2 = 4
   \]
   \[
   \frac{4}{3} = \frac{4}{3}
   \]

3. Now, sum the results:
   \[
   \|\vec{v}\|^2 = 1 + \frac{4}{3}
   \]

4. To add these, express 1 as a fraction with a denominator of 3:
   \[
   1 = \frac{3}{3}
   \]
   \[
   \|\vec{v}\|^2 = \frac{3}{3} + \frac{4}{3} = \frac{7}{3}
   \]

So, the squared magnitude of the vector \(\|\vec{v}\|^2\) is \(\frac{7}{3}\).

If you want to find the magnitude \(\|\vec{v}\|\) itself, you take the square root:

\[
\|\vec{v}\| = \sqrt{\frac{7}{3}} = \frac{\sqrt{7}}{\sqrt{3}} = \frac{\sqrt{21}}{3}
\]

Would you like further details or have any questions?

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Here are some related questions you might find interesting:

1. How do you calculate the magnitude of a vector in 2D space?
2. What is the significance of squaring a vector's magnitude?
3. How do you simplify the square root of a fraction?
4. What is the geometric interpretation of a vector's magnitude?
5. How does changing a vector's components affect its magnitude?

**Tip:** When working with fractions inside a square root, simplify the fraction first if possible before taking the square root.

Learn how to solve for the squared magnitude of a vector using basic arithmetic operations, including squaring numbers and adding fractions. This problem covers key concepts like vector magnitude and square root calculations. Suitable for students in Grades 9-11.

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