https://file.aiquickdraw.com/mathbot/file/172579152411720ve2u2j.jpg

In the image, the vectors \(\vec{a}\) and \(\vec{b}\) are depicted on a coordinate plane, and there are several vector expressions that need to be matched with their respective graphical representations.

Let's break it down:

1. **\(\vec{a} + \vec{b}\)**:
   - This is the vector sum of \(\vec{a}\) and \(\vec{b}\). To find the result, you add the corresponding components of \(\vec{a}\) and \(\vec{b}\). Graphically, this means placing the tail of \(\vec{b}\) at the head of \(\vec{a}\) and drawing a vector from the origin to the new point.
   
2. **\(\vec{a} - \vec{b}\)**:
   - This vector is found by subtracting \(\vec{b}\) from \(\vec{a}\), which is the same as adding \(\vec{a}\) to the negative of \(\vec{b}\). So, reverse the direction of \(\vec{b}\) and then add it to \(\vec{a}\).
   
3. **\(\vec{b} - \vec{a}\)**:
   - Similar to the previous case, but now subtracting \(\vec{a}\) from \(\vec{b}\). This means reversing the direction of \(\vec{a}\) and adding it to \(\vec{b}\).

4. **\(2\vec{b}\)**:
   - This is simply \(\vec{b}\) scaled by 2. Graphically, the direction remains the same, but the magnitude (length) of the vector is doubled.

5. **\(2\vec{a} - \frac{1}{2}\vec{b}\)**:
   - This involves multiplying \(\vec{a}\) by 2 and subtracting half of \(\vec{b}\). Graphically, first scale \(\vec{a}\) by 2, then scale \(\vec{b}\) by \( \frac{1}{2} \) and subtract the resulting vector.

Now, by visually matching the vectors from the expressions to the colored arrows in the image, you can correctly place the labels under their respective vectors. Here’s a rough guide based on common vector conventions:

- The red arrow appears to represent \(\vec{a}\).
- The gray arrow likely represents \(\vec{b}\).
- The other arrows will represent combinations like \(\vec{a} + \vec{b}\), \(2\vec{b}\), etc., based on the rules above.

Do you need further clarification on any of the steps? Here are 5 related questions to deepen your understanding:

1. How do you compute the sum of two vectors algebraically?
2. How does the graphical interpretation of vector addition work?
3. What happens when you multiply a vector by a scalar value?
4. What is the geometric significance of vector subtraction?
5. Can you derive the components of a vector given its initial and terminal points?

**Tip**: When adding or subtracting vectors graphically, always ensure to align the tail of the second vector at the head of the first one for clarity.

In the image, the vectors \(\vec{a}\) and \(\vec{b}\) are depicted on a coordinate plane, and there are several vector expressions that need to be matched with their respective graphical representations.

Explore vector algebra including vector addition, subtraction, and scalar multiplication using graphical methods. Suitable for advanced high school students, this page helps visually decipher complex vector expressions.

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