Find four distinct vectors in F^3 that satisfy x+y+z=0

To find four distinct vectors in \(\mathbb{F}^3\) (where \(\mathbb{F}\) represents a field, typically \(\mathbb{R}\) or \(\mathbb{C}\)) that satisfy the equation \( \mathbf{x} + \mathbf{y} + \mathbf{z} = 0 \), we'll define the vectors in terms of their components \(\mathbf{v} = (v_1, v_2, v_3)\).

The condition \(\mathbf{x} + \mathbf{y} + \mathbf{z} = 0\) translates to:
\[
x_1 + y_1 + z_1 = 0, \quad x_2 + y_2 + z_2 = 0, \quad x_3 + y_3 + z_3 = 0
\]

### Example Vectors:
Let's consider four distinct vectors that satisfy these equations.

1. \(\mathbf{v}_1 = (1, -1, 0)\)
2. \(\mathbf{v}_2 = (-1, 0, 1)\)
3. \(\mathbf{v}_3 = (0, 1, -1)\)
4. \(\mathbf{v}_4 = (1, 1, -2)\)

### Verification:
Now, let's verify that these vectors satisfy the equation \( \mathbf{x} + \mathbf{y} + \mathbf{z} = 0 \).

1. **Vector 1**: \(\mathbf{v}_1 = (1, -1, 0)\)
   - Sum: \(1 + (-1) + 0 = 0\)

2. **Vector 2**: \(\mathbf{v}_2 = (-1, 0, 1)\)
   - Sum: \(-1 + 0 + 1 = 0\)

3. **Vector 3**: \(\mathbf{v}_3 = (0, 1, -1)\)
   - Sum: \(0 + 1 + (-1) = 0\)

4. **Vector 4**: \(\mathbf{v}_4 = (1, 1, -2)\)
   - Sum: \(1 + 1 + (-2) = 0\)

Each vector satisfies the equation, and they are distinct.

Would you like further details on this? Here are some related questions:

1. How can you generalize the condition for any number of vectors?
2. What happens if the field \(\mathbb{F}\) is restricted to only integer values?
3. Can we find a set of vectors that satisfy the equation and are also orthogonal?
4. How does this problem change if we work in \(\mathbb{F}^n\) instead of \(\mathbb{F}^3\)?
5. What is the geometric interpretation of these vectors in \(\mathbb{R}^3\)?

**Tip:** When solving vector problems, always verify each vector meets all given conditions and ensure they are distinct to satisfy the problem constraints.

Learn how to find four distinct vectors in F^3 satisfying the equation x + y + z = 0. This problem explores vectors in three-dimensional space and involves solving linear equations. Suitable for advanced high school and college students.

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