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Chapter 32: Problem 1053

Find the distance from \((6,3,-6)\) to \((10,0,6)\).

Short Answer

Expert verified
The distance between the points A(6, 3, -6) and B(10, 0, 6) is 13 units.

Step by step solution

01

Write down the coordinates of the two points

We are given the coordinates of the two points A(6, 3, -6) and B(10, 0, 6).
02

Recall the distance formula for 3-dimensional space

The distance formula for 3D space is derived from the Pythagorean theorem and is given by: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\) Here, \(d\) is the distance between the points, and \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points.
03

Substitute the given coordinates into the distance formula

We have point A with coordinates \((x_1, y_1, z_1) = (6, 3, -6)\) and point B with coordinates \((x_2, y_2, z_2) = (10, 0, 6)\). Substitute these coordinates into the distance formula. \(d = \sqrt{(10 - 6)^2 + (0 - 3)^2 + (6 - (-6))^2}\)
04

Calculate the distance

Simplify the expression inside the square root. \(d = \sqrt{(4)^2 + (-3)^2 + (12)^2} = \sqrt{16 + 9 + 144}\) Now, add the numbers inside the square root and take the square root of the result. \(d = \sqrt{169} = 13\)
05

Write the final answer

The distance between the points A(6, 3, -6) and B(10, 0, 6) is 13 units.

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