Using distance formula prove that the following points are collinear: `A(4,-3,-1),\ B(5,-7,6)a n d\ C(3,1,-8)`

Explanation

To prove that the points A(4, -3, -1), B(5, -7, 6), and C(3, 1, -8) are collinear using the distance formula, we will follow these steps:

### Step 1: Calculate the distance AB
We use the distance formula for points in three-dimensional space, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

For points A(4, -3, -1) and B(5, -7, 6):
- \(x_1 = 4\), \(y_1 = -3\), \(z_1 = -1\)
- \(x_2 = 5\), \(y_2 = -7\), \(z_2 = 6\)

Substituting these values into the distance formula:
\[ AB = \sqrt{(5 - 4)^2 + (-7 + 3)^2 + (6 + 1)^2} \]
\[ = \sqrt{(1)^2 + (-4)^2 + (7)^2} \]
\[ = \sqrt{1 + 16 + 49} \]
\[ = \sqrt{66} \]

### Step 2: Calculate the distance BC
Now, we calculate the distance between points B(5, -7, 6) and C(3, 1, -8):
- \(x_1 = 5\), \(y_1 = -7\), \(z_1 = 6\)
- \(x_2 = 3\), \(y_2 = 1\), \(z_2 = -8\)

Substituting these values into the distance formula:
\[ BC = \sqrt{(3 - 5)^2 + (1 + 7)^2 + (-8 - 6)^2} \]
\[ = \sqrt{(-2)^2 + (8)^2 + (-14)^2} \]
\[ = \sqrt{4 + 64 + 196} \]
\[ = \sqrt{264} \]

### Step 3: Calculate the distance AC
Next, we calculate the distance between points A(4, -3, -1) and C(3, 1, -8):
- \(x_1 = 4\), \(y_1 = -3\), \(z_1 = -1\)
- \(x_2 = 3\), \(y_2 = 1\), \(z_2 = -8\)

Substituting these values into the distance formula:
\[ AC = \sqrt{(3 - 4)^2 + (1 + 3)^2 + (-8 + 1)^2} \]
\[ = \sqrt{(-1)^2 + (4)^2 + (-7)^2} \]
\[ = \sqrt{1 + 16 + 49} \]
\[ = \sqrt{66} \]

### Step 4: Check for collinearity
For points A, B, and C to be collinear, the sum of the distances AB and AC must equal the distance BC:
\[ AB + AC = \sqrt{66} + \sqrt{66} = 2\sqrt{66} \]
We need to check if \(2\sqrt{66} = \sqrt{264}\):
\[ \sqrt{264} = \sqrt{4 \times 66} = 2\sqrt{66} \]

Since \(AB + AC = BC\), the points A, B, and C are collinear.

### Conclusion
Thus, we have shown that the points A(4, -3, -1), B(5, -7, 6), and C(3, 1, -8) are collinear.

---