Big picture
A map uses two coordinates to locate a place on a flat surface. Locating a classroom fan or a drone also requires height, so three-dimensional coordinate geometry uses ordered triples. Each coordinate records a signed displacement relative to a chosen origin and three mutually perpendicular axes.
The chapter extends familiar plane geometry while preserving the need to interpret coordinates carefully. A point can have a negative coordinate without having a negative distance. Draw the axes, identify which coordinates vanish on a plane and then use algebra to justify the geometry.
Learning goals
Locate points with respect to axes, coordinate planes and octants.
Determine projections and distances from planes and axes.
Derive and apply the three-dimensional distance formula.
Use midpoint and section formulas with the correct ratio order.
Verify geometric properties through distances and coordinates.
Core ideas
1. Coordinate axes and planes
Three mutually perpendicular axes, x, y and z, intersect at the origin, O(0, 0, 0). A point P(x, y, z) is located by signed displacements parallel to these axes. The order is essential: P(2, 3, 5) and Q(5, 3, 2) generally denote different points. Adopt one consistent right-handed axis drawing rather than changing orientation midway through a solution.
Each pair of axes determines a coordinate plane. The xy-plane has z = 0, yz-plane x = 0 and xz-plane y = 0. A point on the x-axis has y = z = 0; analogous conditions apply to the other axes. For example, (−4, 0, 0) lies on the negative x-axis and on both the xy- and xz-planes. A point can therefore belong to more than one coordinate plane.
2. Octants and signs
The three coordinate planes divide space into eight octants. The first octant has x > 0, y > 0 and z > 0. The remaining regions correspond to the other sign combinations. Use a sign table or an explicitly stated convention when octant numbering is requested; the sign combination is the unambiguous description.
Signs of x, y, z | Description |
|---|---|
+, +, + | First octant |
−, +, + | Negative x, positive y and z |
−, −, + | Negative x and y, positive z |
+, −, + | Positive x and z, negative y |
+, +, − | Positive x and y, negative z |
−, +, − | Negative x and z, positive y |
−, −, − | All coordinates negative |
+, −, − | Positive x, negative y and z |
For example, (−2, 5, −7) lies in the region with signs −, +, −. A point with a zero coordinate lies on a boundary plane and is not strictly inside an octant. Never replace zero by a positive sign merely to assign an octant.
3. Projections and plane distances
The perpendicular projection of P(x, y, z) onto the xy-plane is (x, y, 0). Its distance from that plane is |z|. Similarly, distances from the yz- and xz-planes are |x| and |y| respectively. Absolute values ensure nonnegative distances.
For P(−3, 4, −6), projections onto the xy-, yz- and xz-planes are (−3, 4, 0), (0, 4, −6) and (−3, 0, −6). Distances from these planes are 6, 3 and 4 units respectively. Notice that the plane's missing coordinate determines the distance; the xy-plane is not at distance |x| from P. Projection changes only the coordinate perpendicular to the target plane.
4. Distance between two points
For A(x₁, y₁, z₁) and B(x₂, y₂, z₂), three perpendicular displacement components are Δx, Δy and Δz. Apply Pythagoras to the horizontal projection, then to the vertical difference, giving AB = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]. From the origin, OP = √(x² + y² + z²).
Worked example: A(1, −2, 3), B(4, 2, −9). Differences are 3, 4 and −12. Therefore AB² = 3² + 4² + (−12)² = 9 + 16 + 144 = 169, so AB = 13 units. Coordinate order must remain consistent, although reversing all differences gives the same squared distance.
For P(−3, 4, −6), OP = √(9 + 16 + 36) = √61 units. Do not add unsigned coordinate differences directly: a straight-line distance is not generally the same as travelling along three axis-parallel segments.
5. Distance from an axis
The nearest point on the x-axis to P(x, y, z) is (x, 0, 0), so distance from the x-axis is √(y² + z²). Similarly, distances from the y- and z-axes are √(x² + z²) and √(x² + y²). These are different from distances to coordinate planes.
For P(−3, 4, −6), distance from the z-axis is √(9 + 16) = 5 units, while distance from the xy-plane is 6 units. This comparison is a standard application requiring interpretation before formula substitution. A point equidistant from the three coordinate planes satisfies |x| = |y| = |z|, not necessarily x = y = z because signs can differ.
6. Midpoint formula
The midpoint of AB has coordinates ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2). This follows because each coordinate changes halfway between its endpoint values.
For A(−2, 5, 1) and B(6, −1, 9), midpoint M is ((−2 + 6)/2, (5 − 1)/2, (1 + 9)/2) = (2, 2, 5). To find a missing endpoint from A and M, use B = 2M − A coordinate by coordinate. With A(1, 3, −2) and M(4, 1, 5), B = (8 − 1, 2 − 3, 10 + 2) = (7, −1, 12). Substitute the recovered point back into the midpoint formula as an independent check.
7. Internal section formula
If P divides AB internally in ratio AP:PB = m:n, then P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)). The weight attached to the second endpoint is the first ratio term. As m becomes larger, P moves closer to B, consistent with a longer AP segment.
For A(1, 2, −1), B(7, 5, 8) and AP:PB = 2:1, P = ((2 × 7 + 1)/3, (2 × 5 + 2)/3, (2 × 8 − 1)/3) = (5, 4, 5). This is two-thirds of the displacement from A to B. For external division, the corresponding differences replace sums, with denominator m − n, provided m ≠ n; distinguish internal and external wording before using either expression.
8. Ratio and coordinate-plane intersections
A line segment can cross a coordinate plane when the corresponding endpoint coordinates have opposite signs. Let A(2, −1, 3), B(8, 5, −3). The midpoint is (5, 2, 0), so the segment meets the xy-plane halfway. The z-coordinates +3 and −3 cancel equally, establishing AP:PB = 1:1.
For a general ratio problem, set the required coordinate in the section formula equal to zero, solve m:n and then use the same ratio for the other coordinates. If both endpoint z-coordinates are positive, no internal division point lies in the xy-plane. An external intersection may still occur, so geometry can diagnose a negative or otherwise inconsistent ratio before arithmetic is finalised.
9. Geometric proofs and modelling
Distance comparisons can prove equal sides or right angles. For A(0, 0, 0), B(3, 0, 0), C(0, 4, 0), AB² = 9, AC² = 16 and BC² = 25. Since 9 + 16 = 25, the triangle is right-angled at A by the converse of Pythagoras. In genuinely spatial examples the same squared-distance method works without relying on a misleading perspective sketch.
The centroid of a triangle is the average of the three vertex coordinates, obtained by dividing a median in the ratio 2:1. For vertices (0, 0, 0), (6, 0, 0) and (0, 3, 9), the centroid is (2, 1, 3). For a room model, choose one corner as origin and axes along mutually perpendicular edges. The space diagonal of a 6 m × 4 m × 3 m room is √(36 + 16 + 9) = √61 m. A coordinate system is a modelling choice; physical distances remain unchanged if all coordinates are transformed consistently.
A reliable approach to questions
Sketch axes and write the point coordinates in fixed x, y, z order.
Translate geometric conditions into zero coordinates or distance equalities.
Choose distance, midpoint or section formula according to the task.
Keep squared distances until the final step when comparing lengths.
Verify location, ratio and signs against the original geometric condition.
Connections that matter
Plane coordinate geometry supplies the Pythagorean and midpoint foundations.
Physics uses position coordinates and displacement components in space.
Vectors later express section and distance relationships more compactly.
Common mistakes
Using a signed coordinate as a distance can produce an impossible negative length.
Confusing axis and plane distances selects the wrong coordinates.
Reversing section weights puts the point on the wrong part of the segment.
Assigning a boundary point to an octant ignores its zero coordinate.
Revision checklist
I can identify coordinate planes and axis points.
I can describe octants through sign combinations.
I can calculate distances between points and from axes or planes.
I can solve midpoint and section problems in three dimensions.
I can prove simple geometric properties using coordinates.