In Class 10 you fixed the position of a point in a plane with two numbers $(x, y)$. To locate a point in space you need a third measurement — how far the point sits above or below that plane — so every point now carries an ordered triple $(x, y, z)$. The word "ordered" matters: $(2, 3, 5)$ and $(5, 3, 2)$ are completely different points.
The three coordinate axes. Take three mutually perpendicular lines meeting at a single point $O$, the origin. These are the $x$-axis, the $y$-axis and the $z$-axis. Each axis is an ordinary number line — positive on one side of $O$, negative on the other. The three axes are drawn following the right-hand rule: if the fingers of the right hand curl from the positive $x$-axis toward the positive $y$-axis, the thumb points along the positive $z$-axis. This convention keeps everyone's diagrams consistent.
The three coordinate planes. Taken two axes at a time, the axes determine three flat planes that slice space apart. Each plane is named after the two axes lying in it, and on that plane the third coordinate is always zero.
| Plane | Contains axes | Equation | Points on it satisfy |
|---|---|---|---|
| $XY$-plane | $x$- and $y$-axis | $z = 0$ | third coordinate is $0$ |
| $YZ$-plane | $y$- and $z$-axis | $x = 0$ | first coordinate is $0$ |
| $ZX$-plane | $z$- and $x$-axis | $y = 0$ | second coordinate is $0$ |
Reading and locating the coordinates of a point. For a point $P(x, y, z)$, the number $x$ is the perpendicular distance of $P$ from the $YZ$-plane, $y$ the distance from the $ZX$-plane and $z$ the distance from the $XY$-plane — each taken with its proper sign. To plot $P$ you can follow a simple route: start at $O$, walk $x$ units along the $x$-axis, then $y$ units parallel to the $y$-axis, then $z$ units parallel to the $z$-axis. Geometrically $P$ is the far corner of a rectangular box (a cuboid) whose three edges from the origin have lengths $|x|$, $|y|$, $|z|$ along the three axes.
Special positions. A point lies on an axis when its other two coordinates are zero — for example $(5, 0, 0)$ is on the $x$-axis, $(0, -2, 0)$ on the $y$-axis, $(0, 0, 7)$ on the $z$-axis. A point lies on a coordinate plane when exactly one coordinate is zero — $(3, -4, 0)$ on the $XY$-plane, and so on. All three coordinates zero gives the origin itself.
The eight octants. Just as the two axes cut a plane into four quadrants, the three coordinate planes cut space into eight compartments called octants. An octant is fixed entirely by the sign pattern of its points. The first octant is the all-positive region; the standard numbering is given below.
| Octant | $x$ | $y$ | $z$ | Example point |
|---|---|---|---|---|
| I | $+$ | $+$ | $+$ | $(2, 3, 4)$ |
| II | $-$ | $+$ | $+$ | $(-2, 3, 4)$ |
| III | $-$ | $-$ | $+$ | $(-2, -3, 4)$ |
| IV | $+$ | $-$ | $+$ | $(2, -3, 4)$ |
| V | $+$ | $+$ | $-$ | $(2, 3, -4)$ |
| VI | $-$ | $+$ | $-$ | $(-2, 3, -4)$ |
| VII | $-$ | $-$ | $-$ | $(-2, -3, -4)$ |
| VIII | $+$ | $-$ | $-$ | $(2, -3, -4)$ |
Two helpful checks fall out of this table. The top four octants (I–IV) all have $z > 0$ and the bottom four (V–VIII) have $z < 0$, so the sign of $z$ alone tells you whether a point is above or below the $XY$-plane. And octants I–IV, read in the $(x, y)$ signs, are exactly the four quadrants of the $XY$-plane lifted upward.
Deeper Insight — three dimensions as two dimensions plus a height: The whole framework of 3D geometry is built by bolting one more perpendicular axis onto the familiar 2D plane, and almost every formula you meet later is the 2D version with a single extra term tagged on. Notice the symmetry running through the octant table: the first four octants are the four quadrants of the $XY$-plane "lifted" into positive $z$, and octants V–VIII are their mirror images below it — the top four all carry $z > 0$ and the bottom four $z < 0$. This is why a zero coordinate is so informative: one zero pins the point to a coordinate plane, two zeros pin it to an axis, and three zeros give the origin. Hold this "plane plus height" picture firmly and the distance and section formulas in the next two topics will feel like old friends rather than new rules.