Introduction to Three Dimensional Geometry — Study Notes

1. Introduction

The world around us is three dimensional, and mathematics needs coordinates to describe positions in space. While two dimensional geometry uses ordered pairs (x, y), three dimensional geometry uses ordered triples (x, y, z). This chapter introduces the coordinate system in space and the basic tools for measuring distances and locating points.

The chapter begins with the coordinate axes and coordinate planes in space, and how to represent a point by its coordinates. It then develops the distance formula between two points in space and the section formula, including the midpoint. These formulas extend the familiar two dimensional results to three dimensions.

Three dimensional geometry is essential for physics, engineering, computer graphics, and navigation. Understanding how coordinates locate points in space, how distances are measured, and how a point divides a segment prepares students for the more advanced study of lines, planes, and solids in later mathematics.

2. Coordinate Axes and Coordinate Planes in Space

Three mutually perpendicular axes intersect at the origin O: the x-axis, the y-axis, and the z-axis. The positive directions are chosen according to the right-hand rule.

The three coordinate axes determine three coordinate planes:

• XY-plane: the plane containing the x and y axes (z = 0).
• YZ-plane: the plane containing the y and z axes (x = 0).
• ZX-plane: the plane containing the z and x axes (y = 0).

These three planes divide space into eight octants.

Octants

The eight octants have coordinates with definite signs:

• OXYZ: x, y, z all positive.
• O'XYZ: x negative, y, z positive.
• OX'YZ: y negative, x, z positive.
• OXY'Z: z negative, x, y positive.
• and so on for the remaining combinations.

3. Coordinates of a Point in Space

A point in space is represented by an ordered triple (x, y, z). The coordinate x is the distance from the YZ-plane, y the distance from the ZX-plane, and z the distance from the XY-plane.

For example, the point (2, 3, 4) is located 2 units along the x-axis, 3 units along the y-axis, and 4 units along the z-axis.

• A point on the x-axis has coordinates (x, 0, 0).
• A point on the y-axis has coordinates (0, y, 0).
• A point on the z-axis has coordinates (0, 0, z).
• A point in the XY-plane has coordinates (x, y, 0).

4. Distance Between Two Points in Space

The distance between two points P(x1, y1, z1) and Q(x2, y2, z2) in space is:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

This is the three dimensional generalisation of the distance formula in the plane.

Example: The distance between (1, 2, 3) and (4, 6, 3) is sqrt((4-1)^2 + (6-2)^2 + (3-3)^2) = sqrt(9 + 16 + 0) = 5.

Distance from the Origin

The distance of a point P(x, y, z) from the origin O(0, 0, 0) is:

d = sqrt(x^2 + y^2 + z^2)

5. Section Formula in Space

If the point R divides the line segment joining P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n internally, then:

R = ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n))

For external division in the ratio m : n, replace n by -n.

Midpoint

For the midpoint (m : n = 1 : 1), the coordinates are:

R = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)

6. Coordinates of a Point Dividing a Segment with a Given Ratio

Given the coordinates of the end points, the coordinates of the point dividing the segment in the ratio m : n can be found using the section formula. Care must be taken with the signs: internal division uses the formula with both m and n positive, while external division uses -n.

Quick Revision Tables

Table 1: Points on Axes and Planes

Table 2: Key Formulas in 3D Geometry

Mind Map

Important Diagrams (SVG)

Diagram 1: Three Dimensional Coordinate System

Diagram 2: Distance and Section Formula in Space

Common Mistakes

1. Confusing the octant signs. The coordinates of a point in different octants have specific sign patterns; check all three signs.
2. Using the two dimensional distance formula sqrt((x2-x1)^2 + (y2-y1)^2) and forgetting the z-coordinate term.
3. Writing the midpoint as the average of two coordinates only. All three coordinates must be averaged: (x1+x2)/2, (y1+y2)/2, (z1+z2)/2.
4. In the section formula, mixing up m and n: x-coordinate = (m x2 + n x1)/(m + n), with x2 paired with m.
5. Forgetting that external division uses a negative sign for one ratio component.
6. Assuming a point in the XY-plane has coordinate (x, y, z) with nonzero z. In the XY-plane, z = 0.
7. Writing points on the y-axis as (y, 0, 0). A point on the y-axis is (0, y, 0).
8. Confusing which plane contains which two axes, e.g., stating the YZ-plane contains x = 0 axes incorrectly.

Exam Tips

1. Always write the three coordinates in the order (x, y, z) and keep them aligned when applying formulas.
2. To find the distance from the origin, use sqrt(x^2 + y^2 + z^2) directly.
3. In the section formula, first identify which point is P and which is Q and the ratio m : n, then substitute carefully.
4. For midpoint questions, simply average the corresponding coordinates.
5. When a point lies on an axis, two of its coordinates are zero; use this to set up equations for unknown coordinates.
6. Practise locating octants: sign pattern (+, +, +) to (-, -, -) and combinations helps answer octant questions quickly.
7. Check that your computed midpoint actually lies on the line segment and that distances are positive.

Conclusion

Three dimensional geometry extends the coordinate method from the plane to space, providing a systematic way to describe positions using ordered triples. The coordinate axes and planes, the distance formula, and the section formula are the fundamental tools for working with points in space. Understanding how coordinates locate points, how octants are organised, and how segments are divided prepares students for the study of lines, planes, vectors, and solids in higher mathematics. These concepts are also essential for physics, computer graphics, and engineering, where three dimensional thinking is indispensable.