https://r2.erweima.ai/mathbot/image/1725975771399lcy4fs19.jpg

To solve the question, let's analyze which vector among those shown in the image has the largest norm (or magnitude). The norm of a vector is determined by its length. The longer the vector, the larger its norm.

### Observing the Vectors

From the diagram, we have 8 vectors: \( \vec{v_1}, \vec{v_2}, \vec{v_3}, \vec{v_4}, \vec{v_5}, \vec{v_6}, \vec{v_7}, \vec{v_8} \).

#### Checking Their Lengths:

1. **\( \vec{v_1} \)**: Relatively short.
2. **\( \vec{v_2} \)**: Short and oriented in the negative direction of the y-axis.
3. **\( \vec{v_3} \)**: Medium-length vector pointing in the third quadrant.
4. **\( \vec{v_4} \)**: Very short vector along the negative y-axis.
5. **\( \vec{v_5} \)**: Medium-length vector, pointing upwards in the first quadrant.
6. **\( \vec{v_6} \)**: Medium-length vector, directed downward along the negative y-axis.
7. **\( \vec{v_7} \)**: The longest vector, extending towards the negative x-axis.
8. **\( \vec{v_8} \)**: Medium-length vector along the positive x-axis.

### Conclusion:

The vector with the greatest norm is **\( \vec{v_7} \)**, as it is the longest among all the vectors in the diagram.

#### Answer:
The correct choice is:
- **\( \vec{v_7} \)**, because it has the greatest length.

Would you like more details on this, or do you have any questions?

### Additional Related Questions:
1. How do you calculate the norm of a vector given its components?
2. What are the differences between positive and negative projections on the x and y axes?
3. How do you determine the direction of a vector from its components?
4. What is the formula for the dot product of two vectors?
5. How do you find the angle between two vectors using their dot product?

### Tip:
When comparing vector lengths, remember that the magnitude or norm is the Euclidean distance from the origin to the point defined by the vector in a coordinate system.

Parmi les vecteurs suivants, quel est celui qui a la plus grande norme ?

This page helps you understand how to determine which vector has the largest norm in a two-dimensional coordinate system. Learn how to use the vector norm formula and the Pythagorean theorem to solve the problem. Suitable for high school students in Grades 9-12.

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