2D and 3D coordinate geometry

IB Mathematics: Analysis and Approaches SL· 6 min read

1. Core Formulas for 2D and 3D Coordinates★★☆☆☆⏱ 15 min

All core coordinate geometry concepts extend naturally from 2D to 3D by adding a third perpendicular axis, the $z$ -axis. Every point in 3D space is represented by three coordinates $(x, y, z)$ , compared to two coordinates $(x, y)$ in 2D.

📘 Definition

Distance between two points

$P (x_{1}, y_{1}, z_{1}), Q (x_{2}, y_{2}, z_{2})$

The straight-line distance between two points is an extension of Pythagoras' theorem, adding an extra term for the $z$ -coordinate difference in 3D.

Example:

Distance from $(0, 0, 0)$ to $(1, 2, 2)$ is $Distance = 3$

📐 Worked Example

Find the distance between $P (2, - 1, 4)$ and $Q (- 3, 2, 6)$ , and the coordinates of their midpoint.

1
Recall the 3D distance formula:
2
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$
3
Substitute the coordinates of the two points:
4
$d = \sqrt{(-3-2)^2 + (2-(-1))^2 + (6-4)^2} = \sqrt{(-5)^2 + 3^2 + 2^2}$
5
Simplify to get the final distance:
6
$d = \sqrt{25 + 9 + 4} = \sqrt{38} \approx 6.16$
7
The midpoint is the average of each coordinate, so:
8
$M = \left(\frac{2 + (-3)}{2}, \frac{-1 + 2}{2}, \frac{4 + 6}{2}\right) = (-0.5, 0.5, 5)$

2. Common Pitfalls

Wrong move:

Omitting the $z$ -coordinate term when calculating distance in 3D

Why:

Students often forget to extend the 2D distance formula to 3D, leading to incorrect results

Correct move:

Always add the $(z_{2} - z_{1})^{2}$ term when calculating distance between two 3D points

3. Quick Reference Cheatsheet

Concept	2D Formula	3D Formula
Distance between two points	$(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$	$(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$
Midpoint of two points	$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$	$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}, \frac{z _{1} + z _{2}}{2})$

What's Next

The distance and midpoint formulas are foundational tools for geometric problem solving in IB AA SL, appearing in both Paper 1 and Paper 2, often combined with trigonometry to create multi-part exam questions. After mastering these core formulas, you will build on them with triangle trigonometry and trigonometric functions in this unit.