How do you determine the distance between points (1, 2, 3) and (4, 5, 6)?

To determine the distance between points (1, 2, 3) and (4, 5, 6), use the 3D distance formula.

The distance formula in three dimensions is an extension of the Pythagorean theorem. It calculates the straight-line distance between two points in space. The formula is:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Here, \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points. For the points (1, 2, 3) and (4, 5, 6), we substitute the coordinates into the formula:

\[ \text{Distance} = \sqrt{(4 - 1)^2 + (5 - 2)^2 + (6 - 3)^2} \]

First, calculate the differences in each coordinate:

\[ x_2 - x_1 = 4 - 1 = 3 \]
\[ y_2 - y_1 = 5 - 2 = 3 \]
\[ z_2 - z_1 = 6 - 3 = 3 \]

Next, square each of these differences:

\[ (x_2 - x_1)^2 = 3^2 = 9 \]
\[ (y_2 - y_1)^2 = 3^2 = 9 \]
\[ (z_2 - z_1)^2 = 3^2 = 9 \]

Then, add these squared differences together:

\[ 9 + 9 + 9 = 27 \]

Finally, take the square root of the sum:

\[ \sqrt{27} = 3\sqrt{3} \]

So, the distance between the points (1, 2, 3) and (4, 5, 6) is \(3\sqrt{3}\) units.