Find the distance between the points `P(-2, 4, 1) and Q(1, 2, -5).`

Answer

To find the distance between the points \( P(-2, 4, 1) \) and \( Q(1, 2, -5) \), we can use the distance formula in three-dimensional space. The formula for the distance \( d \) between two points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) 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**:
- For point \( P \), the coordinates are \( x_1 = -2 \), \( y_1 = 4 \), \( z_1 = 1 \).
- For point \( Q \), the coordinates are \( x_2 = 1 \), \( y_2 = 2 \), \( z_2 = -5 \).

2. **Substitute the coordinates into the distance formula**:
\[ d = \sqrt{(1 - (-2))^2 + (2 - 4)^2 + (-5 - 1)^2} \]

3. **Calculate each component**:
- Calculate \( (x_2 - x_1) \):
\[ 1 - (-2) = 1 + 2 = 3 \quad \Rightarrow \quad (3)^2 = 9 \]
- Calculate \( (y_2 - y_1) \):
\[ 2 - 4 = -2 \quad \Rightarrow \quad (-2)^2 = 4 \]
- Calculate \( (z_2 - z_1) \):
\[ -5 - 1 = -6 \quad \Rightarrow \quad (-6)^2 = 36 \]

4. **Add the squared differences**:
\[ d = \sqrt{9 + 4 + 36} \]
\[ d = \sqrt{49} \]

5. **Calculate the square root**:
\[ d = 7 \]

### Final Result:
The distance between the points \( P(-2, 4, 1) \) and \( Q(1, 2, -5) \) is \( 7 \) units.

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