Prove by using the distance formula that the points `A (1,2,3), B (-1, -1, -1) and C (3,5,7)` are collinear.

Answer

To prove that the points A(1, 2, 3), B(-1, -1, -1), and C(3, 5, 7) are collinear using the distance formula, we will follow these steps:

### Step-by-Step Solution:

1. **Understand Collinearity**:
- Three points A, B, and C are said to be collinear if the distance between A and B plus the distance between B and C equals the distance between A and C.

2. **Distance Formula in 3D**:
- The distance \(d\) between two points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) in three-dimensional space is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

3. **Calculate Distance AB**:
- For points A(1, 2, 3) and B(-1, -1, -1):
\[ AB = \sqrt{((-1) - 1)^2 + ((-1) - 2)^2 + ((-1) - 3)^2} \]
\[ = \sqrt{(-2)^2 + (-3)^2 + (-4)^2} \]
\[ = \sqrt{4 + 9 + 16} = \sqrt{29} \]

4. **Calculate Distance BC**:
- For points B(-1, -1, -1) and C(3, 5, 7):
\[ BC = \sqrt{(3 - (-1))^2 + (5 - (-1))^2 + (7 - (-1))^2} \]
\[ = \sqrt{(3 + 1)^2 + (5 + 1)^2 + (7 + 1)^2} \]
\[ = \sqrt{4^2 + 6^2 + 8^2} \]
\[ = \sqrt{16 + 36 + 64} = \sqrt{116} = 2\sqrt{29} \]

5. **Calculate Distance AC**:
- For points A(1, 2, 3) and C(3, 5, 7):
\[ AC = \sqrt{(3 - 1)^2 + (5 - 2)^2 + (7 - 3)^2} \]
\[ = \sqrt{(2)^2 + (3)^2 + (4)^2} \]
\[ = \sqrt{4 + 9 + 16} = \sqrt{29} \]

6. **Check Collinearity Condition**:
- Now we check if \(AB + AC = BC\):
\[ AB + AC = \sqrt{29} + \sqrt{29} = 2\sqrt{29} \]
\[ BC = 2\sqrt{29} \]
- Since \(AB + AC = BC\), the points A, B, and C are collinear.

### Conclusion:
Thus, we have proved that the points A(1, 2, 3), B(-1, -1, -1), and C(3, 5, 7) are collinear.