The point (1,2,7) , (2,6,3) , (3,10,-1) are ......... .

Answer

To determine whether the points \( A(1, 2, 7) \), \( B(2, 6, 3) \), and \( C(3, 10, -1) \) are collinear, we can use the concept of distances between the points. If the points are collinear, the distance from \( A \) to \( C \) should equal the sum of the distances from \( A \) to \( B \) and from \( B \) to \( C \).

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A = (1, 2, 7) \)
- Let \( B = (2, 6, 3) \)
- Let \( C = (3, 10, -1) \)

2. **Calculate the Distance \( AB \)**:
The distance formula in three dimensions is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
For points \( A \) and \( B \):
\[ AB = \sqrt{(2 - 1)^2 + (6 - 2)^2 + (3 - 7)^2} \]
\[ = \sqrt{1^2 + 4^2 + (-4)^2} \]
\[ = \sqrt{1 + 16 + 16} \]
\[ = \sqrt{33} \]

3. **Calculate the Distance \( BC \)**:
For points \( B \) and \( C \):
\[ BC = \sqrt{(3 - 2)^2 + (10 - 6)^2 + (-1 - 3)^2} \]
\[ = \sqrt{1^2 + 4^2 + (-4)^2} \]
\[ = \sqrt{1 + 16 + 16} \]
\[ = \sqrt{33} \]

4. **Calculate the Distance \( AC \)**:
For points \( A \) and \( C \):
\[ AC = \sqrt{(3 - 1)^2 + (10 - 2)^2 + (-1 - 7)^2} \]
\[ = \sqrt{2^2 + 8^2 + (-8)^2} \]
\[ = \sqrt{4 + 64 + 64} \]
\[ = \sqrt{132} \]
\[ = 2\sqrt{33} \]

5. **Check the Collinearity Condition**:
For the points to be collinear, the following condition must hold:
\[ AC = AB + BC \]
Substituting the distances we calculated:
\[ 2\sqrt{33} = \sqrt{33} + \sqrt{33} \]
\[ 2\sqrt{33} = 2\sqrt{33} \]

Since the equation holds true, we conclude that the points \( A \), \( B \), and \( C \) are collinear.

### Final Answer:
The points \( (1, 2, 7) \), \( (2, 6, 3) \), and \( (3, 10, -1) \) are collinear.