Find the distance between the following pairs of points :
(i) (-2, 1, -3) and (4, 3, -6)
(ii) (9, -12, -8) and (0,0,0)
(iii) (2,1,-3) and (2, 3, -3)
(iv) (1,0,0) and (4, 4, 5)

Explanation

To find the distance between the given pairs of points in three-dimensional space, we will use the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

where \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points.

### (i) Distance between (-2, 1, -3) and (4, 3, -6)

1. Identify the points:
- \(A(-2, 1, -3)\) and \(B(4, 3, -6)\)
2. Apply the distance formula:
\[ d = \sqrt{(4 - (-2))^2 + (3 - 1)^2 + (-6 - (-3))^2} \]
3. Calculate each term:
- \(4 - (-2) = 4 + 2 = 6\)
- \(3 - 1 = 2\)
- \(-6 - (-3) = -6 + 3 = -3\)
4. Substitute back into the formula:
\[ d = \sqrt{(6)^2 + (2)^2 + (-3)^2} = \sqrt{36 + 4 + 9} = \sqrt{49} = 7 \]

### (ii) Distance between (9, -12, -8) and (0, 0, 0)

1. Identify the points:
- \(A(9, -12, -8)\) and \(B(0, 0, 0)\)
2. Apply the distance formula:
\[ d = \sqrt{(0 - 9)^2 + (0 - (-12))^2 + (0 - (-8))^2} \]
3. Calculate each term:
- \(0 - 9 = -9\)
- \(0 - (-12) = 12\)
- \(0 - (-8) = 8\)
4. Substitute back into the formula:
\[ d = \sqrt{(-9)^2 + (12)^2 + (8)^2} = \sqrt{81 + 144 + 64} = \sqrt{289} = 17 \]

### (iii) Distance between (2, 1, -3) and (2, 3, -3)

1. Identify the points:
- \(A(2, 1, -3)\) and \(B(2, 3, -3)\)
2. Apply the distance formula:
\[ d = \sqrt{(2 - 2)^2 + (3 - 1)^2 + (-3 - (-3))^2} \]
3. Calculate each term:
- \(2 - 2 = 0\)
- \(3 - 1 = 2\)
- \(-3 - (-3) = 0\)
4. Substitute back into the formula:
\[ d = \sqrt{(0)^2 + (2)^2 + (0)^2} = \sqrt{0 + 4 + 0} = \sqrt{4} = 2 \]

### (iv) Distance between (1, 0, 0) and (4, 4, 5)

1. Identify the points:
- \(A(1, 0, 0)\) and \(B(4, 4, 5)\)
2. Apply the distance formula:
\[ d = \sqrt{(4 - 1)^2 + (4 - 0)^2 + (5 - 0)^2} \]
3. Calculate each term:
- \(4 - 1 = 3\)
- \(4 - 0 = 4\)
- \(5 - 0 = 5\)
4. Substitute back into the formula:
\[ d = \sqrt{(3)^2 + (4)^2 + (5)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \]

### Summary of Distances:
1. Distance between (-2, 1, -3) and (4, 3, -6) is **7**.
2. Distance between (9, -12, -8) and (0, 0, 0) is **17**.
3. Distance between (2, 1, -3) and (2, 3, -3) is **2**.
4. Distance between (1, 0, 0) and (4, 4, 5) is **5√2**.