Introduction to Three-Dimensional Geometry - Distance between Two Points

The 3D Cartesian Coordinate System: Space is defined by three mutually perpendicular lines called the $x$, $y$, and $z$ axes that meet at a point called the origin $O(0, 0, 0)$. Visually, this is similar to the corner of a room where the floor meet two walls; the lines of intersection represent the axes, and any point in the room can be located relative to that corner.

Coordinate Planes and Octants: The three axes form three coordinate planes: the $XY$-plane (horizontal), the $YZ$-plane (vertical side), and the $ZX$-plane (vertical front). These planes divide the space into eight regions called octants. Visually, think of a 2D grid being split by a horizontal sheet, resulting in four sections above the sheet and four sections below.

Coordinates of a Point: A point $P$ in 3D space is assigned an ordered triple $(x, y, z)$. The $x$-coordinate is the perpendicular distance from the $YZ$-plane, $y$ is the distance from the $ZX$-plane, and $z$ is the distance from the $XY$-plane. Visually, to reach a point $(2, 3, 4)$, you move 2 units along the $x$-axis, 3 units parallel to the $y$-axis, and 4 units up parallel to the $z$-axis.

The 3D Distance Formula: The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is the length of the line segment $PQ$. It is an extension of the 2D Pythagorean theorem. Visually, the distance $PQ$ represents the length of the longest diagonal of a rectangular box (cuboid) whose edges are parallel to the coordinate axes and have lengths $|x_2 - x_1|$, $|y_2 - y_1|$, and $|z_2 - z_1|$.

Distance from the Origin: A specific application of the distance formula occurs when one point is the origin $O(0, 0, 0)$. The distance of any point $P(x, y, z)$ from the origin is simply the square root of the sum of the squares of its coordinates. Visually, this is the straight-line distance from the very center of the coordinate system to the point in space.

Collinearity in 3D: Three points $A$, $B$, and $C$ are collinear (lie on the same straight line) if the sum of the lengths of any two segments equals the length of the third segment. For example, if $AB + BC = AC$, the points are collinear. This is verified by calculating three separate distances using the distance formula.

Properties of Geometric Figures: The distance formula is used to classify triangles and quadrilaterals in 3D space. For instance, if all three side lengths calculated are equal, the triangle is equilateral; if the sum of the squares of two sides equals the square of the third, it is a right-angled triangle.