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tìm tọa độ điểm M trên mặt phẳng P sao cho tổng 3 vector |MA+MB+MC | đạt giá trị nhỏ nhất

To solve the problem of finding the coordinates of point \( M \) on a plane \( P \) such that the sum of the vectors \( |\vec{MA} + \vec{MB} + \vec{MC}| \) is minimized, we can proceed with the following steps:

### Problem Recap:
- We are given three points \( A \), \( B \), and \( C \) and need to find the point \( M \) on the plane \( P \) such that the vector sum \( |\vec{MA} + \vec{MB} + \vec{MC}| \) is minimized.

### Solution Strategy:
1. **Concept**: The minimum value of the vector sum \( |\vec{MA} + \vec{MB} + \vec{MC}| \) occurs when point \( M \) is the centroid of the triangle formed by points \( A \), \( B \), and \( C \). This is because, at the centroid, the sum of the vectors \( \vec{MA} + \vec{MB} + \vec{MC} \) will result in the smallest possible magnitude due to symmetry.

2. **Finding the Centroid**:
   - If the coordinates of \( A \), \( B \), and \( C \) are given as \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \), then the centroid \( G(x_g, y_g, z_g) \) is given by:
     \[
     x_g = \frac{x_1 + x_2 + x_3}{3}, \quad y_g = \frac{y_1 + y_2 + y_3}{3}, \quad z_g = \frac{z_1 + z_2 + z_3}{3}
     \]
   
3. **Ensuring Point \( M \) Lies on Plane \( P \)**:
   - Once you find the centroid \( G \), you need to ensure it lies on the plane \( P \). If the plane \( P \) is given by an equation of the form \( ax + by + cz + d = 0 \), substitute \( x_g \), \( y_g \), and \( z_g \) into this equation to verify.

4. **Final Step**:
   - If the centroid \( G \) lies on the plane \( P \), then \( M \) coincides with \( G \). If not, you need to project \( G \) onto the plane \( P \) to find \( M \).

If you provide the coordinates of points \( A \), \( B \), and \( C \), and the equation of plane \( P \), I can help you find the exact coordinates of point \( M \).

Would you like to provide more details, or do you need a further explanation on any step?

### Related Questions:
1. How do you derive the formula for the centroid of a triangle in 3D space?
2. What is the significance of the centroid in minimizing vector sums?
3. Can the sum \( |\vec{MA} + \vec{MB} + \vec{MC}| \) be minimized at a point other than the centroid?
4. How do you verify if a point lies on a given plane?
5. What is the geometric interpretation of projecting a point onto a plane?
6. Could the given point \( M \) ever be outside the triangle if it's minimizing the vector sum?
7. How would the solution change if \( A \), \( B \), and \( C \) were not coplanar?
8. What are the implications of having a non-linear plane \( P \) in this context?

### Tip:
Always ensure you check that the centroid lies on the specified plane when solving optimization problems in geometric contexts. This step is crucial to ensuring the validity of your solution.

Learn how to minimize the vector sum |MA + MB + MC| by finding the coordinates of point M on plane P. Understand the geometric concepts of centroids in 3D space and optimization techniques. Suitable for advanced high school students and beyond.

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