Find the distance between the points
(i) (9,-12,-8) and (0,0,0)
(ii) (-3,7,2) and (2,4,-1)
(iii) (-1,3,-4) and (1,-3,4)
(iv) (2,-1,3) and (-2,1,3)

Answer

To find the distance between two points in three-dimensional space, we can 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 the points (9, -12, -8) and (0, 0, 0)

1. Identify the coordinates:
- \(X_1 = 9\), \(Y_1 = -12\), \(Z_1 = -8\)
- \(X_2 = 0\), \(Y_2 = 0\), \(Z_2 = 0\)

2. Substitute into the distance formula:
\[ D = \sqrt{(0 - 9)^2 + (0 - (-12))^2 + (0 - (-8))^2} \]

3. Calculate each term:
- \((0 - 9)^2 = (-9)^2 = 81\)
- \((0 + 12)^2 = 12^2 = 144\)
- \((0 + 8)^2 = 8^2 = 64\)

4. Add the squares:
\[ D = \sqrt{81 + 144 + 64} = \sqrt{289} \]

5. Calculate the square root:
\[ D = 17 \]

### (ii) Distance between the points (-3, 7, 2) and (2, 4, -1)

1. Identify the coordinates:
- \(X_1 = -3\), \(Y_1 = 7\), \(Z_1 = 2\)
- \(X_2 = 2\), \(Y_2 = 4\), \(Z_2 = -1\)

2. Substitute into the distance formula:
\[ D = \sqrt{(2 - (-3))^2 + (4 - 7)^2 + (-1 - 2)^2} \]

3. Calculate each term:
- \((2 + 3)^2 = 5^2 = 25\)
- \((4 - 7)^2 = (-3)^2 = 9\)
- \((-1 - 2)^2 = (-3)^2 = 9\)

4. Add the squares:
\[ D = \sqrt{25 + 9 + 9} = \sqrt{43} \]

### (iii) Distance between the points (-1, 3, -4) and (1, -3, 4)

1. Identify the coordinates:
- \(X_1 = -1\), \(Y_1 = 3\), \(Z_1 = -4\)
- \(X_2 = 1\), \(Y_2 = -3\), \(Z_2 = 4\)

2. Substitute into the distance formula:
\[ D = \sqrt{(1 - (-1))^2 + (-3 - 3)^2 + (4 - (-4))^2} \]

3. Calculate each term:
- \((1 + 1)^2 = 2^2 = 4\)
- \((-3 - 3)^2 = (-6)^2 = 36\)
- \((4 + 4)^2 = 8^2 = 64\)

4. Add the squares:
\[ D = \sqrt{4 + 36 + 64} = \sqrt{104} \]

### (iv) Distance between the points (2, -1, 3) and (-2, 1, 3)

1. Identify the coordinates:
- \(X_1 = 2\), \(Y_1 = -1\), \(Z_1 = 3\)
- \(X_2 = -2\), \(Y_2 = 1\), \(Z_2 = 3\)

2. Substitute into the distance formula:
\[ D = \sqrt{(-2 - 2)^2 + (1 - (-1))^2 + (3 - 3)^2} \]

3. Calculate each term:
- \((-2 - 2)^2 = (-4)^2 = 16\)
- \((1 + 1)^2 = 2^2 = 4\)
- \((3 - 3)^2 = 0^2 = 0\)

4. Add the squares:
\[ D = \sqrt{16 + 4 + 0} = \sqrt{20} \]

### Summary of Distances:
1. Distance between (9, -12, -8) and (0, 0, 0) is **17 units**.
2. Distance between (-3, 7, 2) and (2, 4, -1) is **\(\sqrt{43}\) units**.
3. Distance between (-1, 3, -4) and (1, -3, 4) is **\(\sqrt{104}\) units**.
4. Distance between (2, -1, 3) and (-2, 1, 3) is **\(\sqrt{20}\) units**.