3D Geometry Essentials

3D geometry extends 2D concepts into three dimensions, using coordinates \((x, y, z)\) to describe points in space.

Coordinates

A point in 3D is \((x, y, z)\), where \(x\) is left-right, \(y\) is forward-back, and \(z\) is up-down.

Distance between \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\):

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

Example: From (1, 2, 3) to (4, 6, 8):

\[ d = \sqrt{(4-1)^2 + (6-2)^2 + (8-3)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \approx 7.07 \]

3D geometry is used in computer graphics (e.g., rendering objects), physics (e.g., vector motion), and architecture (e.g., structural design).