Question

Find the distance between the points. \((2,2,3), \quad(4,-5,6)\)

Short answer

The distance between the points \( (2,2,3) \) and \( (4,-5,6) \) is \( \sqrt{62} \).

Step-by-step solution

Identify Points in Space

The given two points in the three-dimensional space are \( (2,2,3) \) and \( (4,-5,6) \). Let's label them as \( P1(x_1, y_1, z_1) = (2,2,3) \) and \( P2(x_2, y_2, z_2) = (4,-5,6) \).

Apply the Euclidean Distance Formula

The Euclidean distance between two points \( P1(x_1, y_1, z_1) \) and \( P2(x_2, y_2, z_2) \) in a three-dimensional space is given by the formula \( D = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}} \). Let's substitute the coordinates into the distance formula.

Substitute the coordinates

Substituting the coordinates, we have \( D = \sqrt{{(4 - 2)^2 + ((-5) - 2)^2 + (6 - 3)^2}} \).

Simplify the Expression

After simplifying the above expression, we get \( D = \sqrt{{2^2 + (-7)^2 + 3^2}} = \sqrt{{4 + 49 + 9}} = \sqrt{62} \).

Final Answer

So, the distance between the given two points is \( \sqrt{62} \).