Question

Find the distance between the two points. $$ (-1,-5,7),(-3,4,-4) $$

Short answer

The distance between the points (-1,-5,7) and (-3,4,-4) is \(\sqrt{206}\)

Step-by-step solution

Identify the coordinates

Identify the coordinates of the two points. Here, the coordinates for the first point are (-1,-5,7) and for the second point are (-3,4,-4).

Apply the 3D distance formula

The 3-dimensional distance formula is: \[d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\] Substituting the given coordinates into the formula, we get: \[ d = \sqrt{(-3-(-1))^2 + (4-(-5))^2 + (-4-7)^2} \]

Simplifying the equation

The next step is to simplify the equation: \[d = \sqrt{(-2)^2 + 9^2 + (-11)^2} = \sqrt{4 + 81 + 121} \]

Final calculation

Finally, we perform the arithmetic operations: \[d = \sqrt{206} \]