Unit 9: Introduction to 3D Geometry
Contents
1. Coordinate Axes and Planes
In three-dimensional space, we use three mutually perpendicular lines passing through a common origin O to define position. These are the X, Y, and Z axes.
The Coordinate Planes
These axes taken in pairs determine three coordinate planes:
- XY-plane: Contains X and Y axes; z = 0 for all points here.
- YZ-plane: Contains Y and Z axes; x = 0 for all points here.
- ZX-plane: Contains Z and X axes; y = 0 for all points here.
2. Coordinates and Octants
The three coordinate planes divide the entire space into eight parts known as Octants. The coordinates of a point P are written as an ordered triple (x, y, z).
| Octant | I | II | III | IV | V | VI | VII | VIII |
|---|---|---|---|---|---|---|---|---|
| x | + | - | - | + | + | - | - | + |
| y | + | + | - | - | + | + | - | - |
| z | + | + | + | + | - | - | - | - |
3. Distance Between Two Points
The distance between two points P(x1, y1, z1) and Q(x2, y2, z2) is an extension of the 2D distance formula.
4. Section Formula
The section formula helps find the coordinates of a point R that divides the line segment joining P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n.
Internal Division:
External Division:
Mid-point Formula: If R is the mid-point (ratio 1:1), then:
R = ( (x1 + x2) / (2), (y1 + y2) / (2), (z1 + z2) / (2) )
5. Exam Focus Enhancements
- Centroid of a Triangle: If asked for the centroid of a triangle with vertices (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), the formula is:
G = ( (x1+x2+x3) / (3), (y1+y2+y3) / (3), (z1+z2+z3) / (3) ). - The "k:1" Method: To find the ratio in which a point divides a line, always assume the ratio is k : 1. If k is positive, it's internal; if negative, it's external.
- Points on Axes: A point on the X-axis is (x, 0, 0); on the Y-axis is (0, y, 0); on the Z-axis is (0, 0, z).
- Signs in Octants: Students often confuse the 4th and 8th octants. Note that z is positive for the first four and negative for the last four.
- External Division Formula: Forgetting the minus sign in the denominator (m-n) often leads to calculation errors.
- Coordinate Planes: Misidentifying the plane. Remember: in the XY-plane, the Z-coordinate is zero.
Q: What is the distance of point (x, y, z) from the origin?
A: d = √(x2 + y2 + z2).
Q: How many octants are there?
A: There are 8 octants, just as there are 4 quadrants in 2D geometry.