Three-Dimensional Coordinate System
The three-dimensional coordinate system extends the 2D plane into 3D space by adding a third axis, typically labeled z. Points in 3D space are represented by ordered triples $(x, y, z)$.
Distance Between Two Points in 3D Space
The distance formula in 3D is an extension of the 2D Pythagorean theorem:
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$
Where $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are the coordinates of the two points.
ExampleCalculate the distance between points A(1, 2, 3) and B(4, 6, 8):
$d = \sqrt{(4-1)^2 + (6-2)^2 + (8-3)^2}$ $= \sqrt{3^2 + 4^2 + 5^2}$ $= \sqrt{9 + 16 + 25}$ $= \sqrt{50} = 5\sqrt{2}$
Midpoint in 3D Space
The midpoint formula in 3D is a straightforward extension of the 2D version:
$M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})$
ExampleFind the midpoint of the line segment connecting A(1, 2, 3) and B(4, 6, 8):
$M = (\frac{1+4}{2}, \frac{2+6}{2}, \frac{3+8}{2}) = (2.5, 4, 5.5)$
Volume and Surface Area of 3D Solids
Right Pyramid
A right pyramid has a polygonal base and triangular faces that meet at a point (apex) directly above the center of the base.
Volume: $V = \frac{1}{3} \times B \times h$, where B is the area of the base and h is the height.
For a square pyramid with base side 6 units and height 8 units: $V = \frac{1}{3} \times 6^2 \times 8 = 96$ cubic units
