Class 11: Introduction to 3D Geometry
Coordinate Axes, Planes, Octants & Distance Formula
1. Coordinate Axes and Planes
| Coordinate Axes | Three mutually perpendicular lines intersecting at Origin $$O$$. X-axis, Y-axis, Z-axis |
| Coordinate Planes | XY-Plane: Determined by X and Y axes ($$z=0$$) YZ-Plane: Determined by Y and Z axes ($$x=0$$) ZX-Plane: Determined by Z and X axes ($$y=0$$) |
| Octants | The three coordinate planes divide the space into eight parts called Octants. |
2. Coordinates of a Point
A point $$P$$is represented by an ordered triplet$$(x, y, z)$$
| Location | Coordinates Form |
|---|---|
| Point on X-axis | $$(x, 0, 0)$$ |
| Point on Y-axis | $$(0, y, 0)$$ |
| Point on Z-axis | $$(0, 0, z)$$ |
| Point in XY-plane | $$(x, y, 0)$$ |
Sign Convention in Octants:
| Octant | I | II | III | IV | V | VI | VII | VIII |
|---|---|---|---|---|---|---|---|---|
| Sign | +,+,+ | -,+,+ | -,-,+ | +,-,+ | +,+,- | -,+,- | -,-,- | +,-,- |
3. Distance Formula
Distance between two points $$P(x_1, y_1, z_1)$$and$$Q(x_2, y_2, z_2)$$
| Distance Formula | $$PQ = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}$$ |
| Distance from Origin | Distance of $$P(x, y, z)$$from Origin$$O(0,0,0)$$ $$OP = \sqrt{x^2 + y^2 + z^2}$$ |
| Collinear Points | Three points A, B, C are collinear if sum of any two distances equals the third distance (e.g., $$AB + BC = AC$$). |