Calculate the distance between two points `(0, -1, 1) and (3, 3, 13)`.

Explanation

To calculate the distance between two points \((0, -1, 1)\) and \((3, 3, 13)\), we can use the distance formula for three-dimensional space. The distance \(d\) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by:

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

Let's apply this formula step by step.

1. **Identify the coordinates of the points:**
- Point \(P\): \((0, -1, 1)\)
- \(x_1 = 0\)
- \(y_1 = -1\)
- \(z_1 = 1\)
- Point \(Q\): \((3, 3, 13)\)
- \(x_2 = 3\)
- \(y_2 = 3\)
- \(z_2 = 13\)

2. **Substitute the coordinates into the distance formula:**
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
\[ d = \sqrt{(3 - 0)^2 + (3 - (-1))^2 + (13 - 1)^2} \]

3. **Calculate the differences:**
\[ x_2 - x_1 = 3 - 0 = 3 \]
\[ y_2 - y_1 = 3 - (-1) = 3 + 1 = 4 \]
\[ z_2 - z_1 = 13 - 1 = 12 \]

4. **Square each difference:**
\[ (x_2 - x_1)^2 = 3^2 = 9 \]
\[ (y_2 - y_1)^2 = 4^2 = 16 \]
\[ (z_2 - z_1)^2 = 12^2 = 144 \]

5. **Sum the squares:**
\[ 9 + 16 + 144 = 169 \]

6. **Take the square root of the sum:**
\[ d = \sqrt{169} = 13 \]

Therefore, the distance between the points \((0, -1, 1)\) and \((3, 3, 13)\) is \(13\) units.