3D world

In three-dimensional space, a point is determined by three variables:

(x, y, z).

The coordinate axes give us three independent directions, so one number is no longer enough and two numbers are also not enough. We need all three coordinates to locate a point in 3D space.

Picking a Point in 3D

x =y =z =

Point: (2, 1, 3)

Magnitude: 3.74

💡 Tip: in vector mode, drag inside the graph to move the endpoint. Hold Shift while dragging to adjust z.

We use a right-handed 3D coordinate system.

Point the index finger of your right hand along the positive $x$ -axis.
Point the middle finger along the positive $y$ -axis.
Then the thumb points along the positive $z$ -axis.

Equivalently, if you curl the fingers of your right hand from the $x$ -axis toward the $y$ -axis, your thumb gives the positive orientation of the $z$ -axis.

The definition of implicit function

An explicit function writes one variable directly in terms of another, such as

y = f(x).

A parametric function uses a parameter and writes both coordinates as functions of that parameter:

x = x(t), \quad y = y(t).

An implicit function describes a relationship by an equation involving both variables without solving for one variable explicitly:

F(x,y) = 0.

The three viewpoints are often compared like this:

Type	Form	Example	What it describes
Explicit function	$y=f(x)$	$y=x^2$	one output $y$ for each input $x$
Implicit function	$F(x,y)=0$	$x^2+y^2-1=0$	a relation or curve in the plane
Parametric function	$x=x(t),\ y=y(t)$	$x=\cos t,\ y=\sin t$	a curve traced by a parameter

The circle is a simple example of an implicit curve:

x^2 + y^2 - 1 = 0.

Unit circle as an implicit curve

Distance in 3D

For two points

P_1 = (x_1, y_1, z_1), \quad P_2 = (x_2, y_2, z_2),

the distance is

d(P_1,P_2) = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}.

This formula comes from applying the Pythagorean theorem twice. First, the horizontal displacement in the $xy$ -plane gives one right triangle. Then the vertical displacement in $z$ gives a second right triangle, and together they form a cuboid.

Distance Between Two Points in 3D

x₁ =y₁ =z₁ =

x₂ =y₂ =z₂ =

Point A: (1, 1, 1)

Point B: (4, 3, 5)

Distance AB: 5.39

💡 Tip: in vector mode, drag inside the graph to move the endpoint. Hold Shift while dragging to adjust z.

For the two points above, the distance is

\sqrt{(4-1)^2 + (3-1)^2 + (5-1)^2} = \sqrt{9 + 4 + 16} = \sqrt{29}.

Sphere in 3D

The general equation of a sphere in 3D is

(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2,

where $(a,b,c)$ is the center and $r$ is the radius.

Now consider the equation

x^2 + y^2 + z^2 + 4x - 6y + 2z + 6 = 0.

We complete the squares:

\begin{aligned} x^2 + 4x &= (x+2)^2 - 4, \\ y^2 - 6y &= (y-3)^2 - 9, \\ z^2 + 2z &= (z+1)^2 - 1. \end{aligned}

Substituting these into the equation gives

(x+2)^2 + (y-3)^2 + (z+1)^2 - 4 - 9 - 1 + 6 = 0,

(x+2)^2 + (y-3)^2 + (z+1)^2 = 8.

Therefore this is a sphere with center

(-2, 3, -1)

and radius

\sqrt{8} = 2\sqrt{2}.

3D Coordinate