A Flying Drone's Position Needs Three Numbers, Not Two

On a flat sheet, two numbers $(x,y)$ fix any point. But a drone hovering in a room also has a height, so it needs a third number. Adding a third axis, perpendicular to the other two, gives coordinates $(x,y,z)$ for every point in space.

The ideas of plane coordinate geometry carry over with one extra term.

This chapter covers the axes, planes and octants, locating points, the distance formula, and collinearity and triangles in space.

**Three mutually perpendicular lines through the origin $O$ form the x-, y- and z-axes; each pair of axes fixes a coordinate plane, and the three planes divide space into $2^3=8$ octants.

The coordinate planes:

- XY-plane** — contains the x- and y-axes; every point on it has $z=0$
- YZ-plane — contains the y- and z-axes; every point on it has $x=0$
- ZX-plane — contains the z- and x-axes; every point on it has $y=0$

Octants. Each plane splits space into two halves, so three planes make $2\times 2\times 2=8$ regions, numbered I to VIII.

An everyday example. Stand in the corner of a classroom: the two edges along the floor are like the x- and y-axes, the vertical edge up the wall is the z-axis, and the floor itself is the XY-plane.

The substance. The axes are arranged in a right-handed system: curling the fingers of the right hand from the x-axis towards the y-axis points the thumb along the positive z-axis.

**To locate $(x,y,z)$, move $x$ units along the x-axis, then $y$ units parallel to the y-axis, then $z$ units parallel to the z-axis; the signs of the three coordinates identify the octant.

Signs in each octant** $(x,y,z)$:

- I $(+,+,+)$ II $(-,+,+)$ III $(-,-,+)$ IV $(+,-,+)$
- V $(+,+,-)$ VI $(-,+,-)$ VII $(-,-,-)$ VIII $(+,-,-)$

Worked examples.

- $(2,-3,4)$ — signs $(+,-,+)$: octant IV
- $(-1,-2,-3)$ — all negative: octant VII
- $(4,0,-5)$ — $y=0$, so it lies in the ZX-plane, not inside any octant

Distances from planes and axes. For $P(2,-3,4)$:

$$\text{distance from XY-plane}=|z|=4\text{distance from x-axis}=\sqrt{y^2+z^2}=\sqrt{9+16}=5$$

An everyday example. A ceiling fan in a room can be located by how far it is from two walls and how high it is above the floor.

The boundary case. A point with any coordinate equal to zero lies on a coordinate plane, and one with two zeros lies on an axis.

**The distance between $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ is $PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$, which comes from applying Pythagoras twice.

Why.** The horizontal distance between the points satisfies $d^2=(x_2-x_1{)}^2+(y_2-y_1{)}^2$. The vertical gap is $z_2-z_1$, and a second right triangle gives $P{Q}^2=d^2+(z_2-z_1{)}^2$.

Worked example 1. Distance between $(1,-3,4)$ and $(-4,1,2)$:

$$PQ=\sqrt{(-5{)}^2+4^2+(-2{)}^2}=\sqrt{25+16+4}=\sqrt{45}=3\sqrt{5}$$

Worked example 2 — a room. A hall is $12$ m long, $4$ m wide and $3$ m high. Its longest straight line, from one floor corner to the opposite ceiling corner, is

$$\sqrt{12^2+4^2+3^2}=\sqrt{144+16+9}=\sqrt{169}=13\text{ m}$$

Worked example 3 — from the origin. $(2,-3,6)$ is $\sqrt{4+9+36}=7$ units from $O$.

An everyday example. **Checking whether a $13$ m bamboo pole fits inside a hall is exactly the second example.

The substance. Setting $z_1=z_2$ gives back the plane distance formula**, so this is its natural extension.

Three points are collinear when the two shorter distances add up exactly to the longest; otherwise they form a triangle, which is classified by comparing its side lengths and checking Pythagoras.

Worked example 1 — collinear. $P(-2,3,5)$, $Q(1,2,3)$, $R(7,0,-1)$.

$$PQ=\sqrt{9+1+4}=\sqrt{14}QR=\sqrt{36+4+16}=2\sqrt{14}PR=\sqrt{81+9+36}=3\sqrt{14}$$

Since $\sqrt{14}+2\sqrt{14}=3\sqrt{14}$, the points are collinear.

Worked example 2 — a triangle. $A(0,7,-10)$, $B(1,6,-6)$, $C(4,9,-6)$.

$$AB=\sqrt{1+1+16}=\sqrt{18}BC=\sqrt{9+9+0}=\sqrt{18}AC=\sqrt{16+4+16}=6$$

$AB=BC$, so the triangle is isosceles, and $A{B}^2+B{C}^2=36=A{C}^2$, so it is also **right-angled at $B$.

An everyday example. Three drones at a light show appear in a straight line only if their distances satisfy this test.

The substance. Equal-looking decimals are not enough** — keep surds such as $\sqrt{14}$ exact so the sum can be checked precisely.