https://r2.erweima.ai/mathbot/image/1725422865293b3kp0sh0.jpg

The expression in the new image is:

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

Let's solve this step by step:

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

2. Next, calculate \(\left(\frac{2}{3}\right)^2\):
   \[
   \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}
   \]

3. Now, sum the results:
   \[
   \|\vec{v}\|^2 = 1 + \frac{4}{9}
   \]

4. To add these, express 1 as a fraction with a denominator of 9:
   \[
   1 = \frac{9}{9}
   \]
   \[
   \|\vec{v}\|^2 = \frac{9}{9} + \frac{4}{9} = \frac{13}{9}
   \]

So, the squared magnitude of the vector \(\|\vec{v}\|^2\) is \(\frac{13}{9}\).

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

\[
\|\vec{v}\| = \sqrt{\frac{13}{9}} = \frac{\sqrt{13}}{3}
\]

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 convert a fraction to a decimal, and vice versa?
2. What is the process for adding fractions with different denominators?
3. How do you interpret the square of a fraction geometrically?
4. What does the magnitude of a vector represent in physical terms?
5. How does squaring affect negative numbers in mathematical operations?

**Tip:** When dealing with fractions, always check if the fraction can be simplified before performing operations like squaring or taking square roots.

Solve the expression: ||v||^2 = -1^2 + (2/3)^2

This guide provides a detailed solution to the vector magnitude problem ||v||^2 = -1^2 + (2/3)^2, covering algebraic operations and the application of the Pythagorean theorem. Suitable for students in Grades 9-11.

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