Introduction to Three-Dimensional Geometry - Coordinates of a Point

Coordinate Axes and Origin: In 3D geometry, we use three mutually perpendicular lines called the $X$, $Y$, and $Z$ axes that intersect at a single point called the Origin $O(0, 0, 0)$. Visually, these axes can be imagined as the lines formed where two walls and the floor of a room meet.

Coordinate Planes: The three axes taken in pairs determine three coordinate planes: the $XY$-plane (where $z=0$), the $YZ$-plane (where $x=0$), and the $ZX$-plane (where $y=0$). These planes are like the flat surfaces of the walls and floor in a 3D space.

Octants: The three coordinate planes divide the entire space into eight regions known as octants. The signs of the coordinates $(x, y, z)$ of a point vary depending on its octant; for instance, in the first octant, all coordinates are positive $(+, +, +)$, while in the eighth octant, they are $(+, -, -)$.

Coordinates of a Point: A point $P$ in space is represented by an ordered triple $(x, y, z)$. The value of $x$ is the signed perpendicular distance from the $YZ$-plane, $y$ from the $XZ$-plane, and $z$ from the $XY$-plane. Visually, moving from the origin to $P$ involves traveling along the $x$-axis, then parallel to the $y$-axis, and finally parallel to the $z$-axis.

Distance from Axes: The perpendicular distance of a point $P(x, y, z)$ from the $X$-axis is given by $\sqrt{y^2 + z^2}$, from the $Y$-axis by $\sqrt{x^2 + z^2}$, and from the $Z$-axis by $\sqrt{x^2 + y^2}$.

Section Formula Concept: This describes the position of a point that divides a line segment joining two points $P$ and $Q$. If the point is between $P$ and $Q$, it is internal division; if it lies on the extension of the line segment, it is external division.

Centroid of a Triangle: The centroid is the point where the medians of a triangle meet in 3D space. It is found by taking the arithmetic mean of the $x$, $y$, and $z$ coordinates of the three vertices.