Find the distance between the points A(-2,1,3) and B(1,2,6).

Explanation

To find the distance between the points A(-2, 1, 3) and B(1, 2, 6), we can use the distance formula for three-dimensional space. The formula is given by:

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

### Step 1: Identify the coordinates
We have the coordinates of the points:
- Point A: \( A(x_1, y_1, z_1) = A(-2, 1, 3) \)
- Point B: \( B(x_2, y_2, z_2) = B(1, 2, 6) \)

### Step 2: Substitute the coordinates into the formula
Now, we will substitute the coordinates into the distance formula:

\[ d = \sqrt{(1 - (-2))^2 + (2 - 1)^2 + (6 - 3)^2} \]

### Step 3: Simplify each term
Now, simplify each term inside the square root:

1. For the x-coordinates:
\[ 1 - (-2) = 1 + 2 = 3 \quad \Rightarrow \quad (3)^2 = 9 \]

2. For the y-coordinates:
\[ 2 - 1 = 1 \quad \Rightarrow \quad (1)^2 = 1 \]

3. For the z-coordinates:
\[ 6 - 3 = 3 \quad \Rightarrow \quad (3)^2 = 9 \]

### Step 4: Add the squared terms
Now, we add the squared terms together:

\[ d = \sqrt{9 + 1 + 9} = \sqrt{19} \]

### Step 5: Final result
Thus, the distance between the points A and B is:

\[ d = \sqrt{19} \]