To calculate the distance between the two points \( A(0, -1, 1) \) and \( B(3, 3, 13) \), we can use the distance formula in three-dimensional space. The distance \( d \) between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]
### Step-by-step Solution:
1. **Identify the coordinates of the points**:
- Point \( A \) has coordinates \( (x_1, y_1, z_1) = (0, -1, 1) \)
- Point \( B \) has coordinates \( (x_2, y_2, z_2) = (3, 3, 13) \)
2. **Substitute the coordinates into the distance formula**:
\[
d = \sqrt{(3 - 0)^2 + (3 - (-1))^2 + (13 - 1)^2}
\]
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 the differences**:
\[
d = \sqrt{(3)^2 + (4)^2 + (12)^2}
\]
- \( (3)^2 = 9 \)
- \( (4)^2 = 16 \)
- \( (12)^2 = 144 \)
5. **Add the squared differences**:
\[
d = \sqrt{9 + 16 + 144}
\]
\[
d = \sqrt{169}
\]
6. **Take the square root**:
\[
d = 13
\]
### Final Answer:
The distance between the two points \( A(0, -1, 1) \) and \( B(3, 3, 13) \) is \( 13 \) units.
