Question

Find the distance between the two points. $$ (-4,-1,1),(2,-1,5) $$

Short answer

The distance between the two points is \( \sqrt{52} \) or approximately 7.21 units.

Step-by-step solution

Identify the coordinates

The given coordinates are (x1, y1, z1) = (-4, -1, 1) and (x2, y2, z2) = (2, -1, 5).

Substitute in the distance formula

Substitute these values into the distance formula: \( \sqrt{(x_{2} - x_{1})^2 + (y_{2} - y_{1})^2 + (z_{2} - z_{1})^2} \) = \( \sqrt{(2 - (-4))^2 + ((-1) - (-1))^2 + (5 - 1)^2} \)

Calculate the distance

Perform the operation within each parentheses and then square the results: \( \sqrt{(2 + 4)^2 + (0)^2 + (5 - 1)^2} = \sqrt{(6)^2 + (0)^2 + (4)^2} = \sqrt{36 + 0 + 16} = \sqrt{52} \)