The Central Idea
A point in three-dimensional Cartesian space is represented by an ordered triple
\[ \boxed{(x,y,z)}. \]The three coordinates describe position along three mutually perpendicular directions:
- \(x\): position along the \(x\)-axis;
- \(y\): position along the \(y\)-axis;
- \(z\): position along the \(z\)-axis.
The order matters just as it does with ordered pairs.
From the Plane to Space
In two dimensions, the coordinate plane uses
\[ (x,y). \]Adding a third perpendicular coordinate produces
\[ (x,y,z). \]The set of all ordered triples of real numbers is written
\[ \boxed{\mathbb{R}^3=\mathbb{R}\times\mathbb{R}\times\mathbb{R}}. \]The Three Coordinate Axes
Three-dimensional Cartesian space contains three coordinate axes:
- the \(x\)-axis;
- the \(y\)-axis;
- the \(z\)-axis.
They intersect at one point and are pairwise perpendicular.
The Origin
The three coordinate axes intersect at the origin
\[ \boxed{O=(0,0,0)}. \]This point is the reference location from which all three coordinates are measured.
The \(x\)-Axis in Three Dimensions
Every point on the \(x\)-axis has
\[ y=0 \]and
\[ z=0. \]Therefore the \(x\)-axis can be described as
\[ \boxed{(x,0,0)}. \]The \(y\)-Axis in Three Dimensions
Every point on the \(y\)-axis has
\[ x=0,\qquad z=0. \]Thus the \(y\)-axis consists of points
\[ \boxed{(0,y,0)}. \]The \(z\)-Axis
Every point on the \(z\)-axis has
\[ x=0,\qquad y=0. \]Therefore the \(z\)-axis is described by
\[ \boxed{(0,0,z)}. \]The Coordinate Planes
Every pair of coordinate axes determines a coordinate plane.
The three coordinate planes are:
\[ \boxed{xy\text{-plane}: z=0}, \] \[ \boxed{xz\text{-plane}: y=0}, \] \[ \boxed{yz\text{-plane}: x=0}. \]Each coordinate plane is a two-dimensional copy of \(\mathbb{R}^2\) embedded inside \(\mathbb{R}^3\).
Reading an Ordered Triple
Consider
\[ P=(3,-2,5). \]Its coordinates say:
- \(x=3\);
- \(y=-2\);
- \(z=5\).
The point is located by combining all three signed coordinate displacements.
Plotting a Point in Space
To plot
\[ (4,2,3), \]conceptually begin at the origin and move according to the three coordinates.
One convenient interpretation is:
- locate \((4,2,0)\) in the \(xy\)-plane;
- move \(3\) units parallel to the \(z\)-axis.
The final point is \((4,2,3)\).
Negative Coordinates in Space
Negative coordinates indicate motion in the negative direction of the corresponding axis.
For
\[ (-3,4,-2), \]the point lies in the negative \(x\)-direction, positive \(y\)-direction, and negative \(z\)-direction from the origin.
Octants
The three coordinate planes divide space into eight regions called octants.
Each octant corresponds to one of the eight possible sign combinations of
\[ (x,y,z). \]The first octant is conventionally
\[ \boxed{x>0,\qquad y>0,\qquad z>0}. \]Octant Naming Is Less Standard Than Quadrant Naming
In two dimensions, Quadrants I through IV have a widely standardized numbering convention.
In three dimensions, only the first octant is universally named in a consistent way.
For the other seven regions, sign descriptions such as
\[ (-,+,-) \]are often clearer than relying on an octant number.
Points on Coordinate Planes
A point lies on a coordinate plane when one coordinate is zero.
For example,
\[ (3,-5,0) \]lies on the \(xy\)-plane because
\[ z=0. \]Likewise,
\[ (0,4,7) \]lies on the \(yz\)-plane.
Points on Coordinate Axes
A point lies on a coordinate axis when two of its coordinates are zero.
For example,
\[ (0,-6,0) \]lies on the \(y\)-axis.
Projection onto the \(xy\)-Plane
To project
\[ (x,y,z) \]onto the \(xy\)-plane, set
\[ z=0. \]Thus,
\[ \boxed{(x,y,z)\mapsto(x,y,0)}. \]This removes the height component while preserving horizontal location in the \(xy\)-plane.
Projection onto the \(xz\)-Plane
Projection onto the \(xz\)-plane sets
\[ y=0. \]Therefore,
\[ \boxed{(x,y,z)\mapsto(x,0,z)}. \]Projection onto the \(yz\)-Plane
Projection onto the \(yz\)-plane sets
\[ x=0. \]Thus,
\[ \boxed{(x,y,z)\mapsto(0,y,z)}. \]Distance from the Origin
For
\[ P=(x,y,z), \]the distance from the origin is
\[ \boxed{ OP=\sqrt{x^2+y^2+z^2} }. \]This is the three-dimensional extension of
\[ \sqrt{x^2+y^2} \]in the plane.
Deriving the Three-Dimensional Distance Formula
Let
\[ P=(x_1,y_1,z_1) \]and
\[ Q=(x_2,y_2,z_2). \]The coordinate differences are
\[ \Delta x=x_2-x_1, \] \[ \Delta y=y_2-y_1, \] \[ \Delta z=z_2-z_1. \]Applying the Pythagorean relationship twice gives
\[ d^2=(\Delta x)^2+(\Delta y)^2+(\Delta z)^2. \]Three-Dimensional Distance Formula
Therefore,
\[ \boxed{ d= \sqrt{ (x_2-x_1)^2+ (y_2-y_1)^2+ (z_2-z_1)^2 } }. \]Distance remains nonnegative and does not depend on which endpoint is listed first.
Example: Distance in Space
Find the distance between
\[ A=(1,2,3) \]and
\[ B=(5,-1,9). \]Then
\[ \Delta x=4,\qquad \Delta y=-3,\qquad \Delta z=6. \]Therefore,
\[ d= \sqrt{4^2+(-3)^2+6^2} = \sqrt{61}. \]Thus,
\[ \boxed{AB=\sqrt{61}}. \]Midpoint in Three Dimensions
The midpoint formula extends coordinate by coordinate.
For
\[ P=(x_1,y_1,z_1) \]and
\[ Q=(x_2,y_2,z_2), \]the midpoint is
\[ \boxed{ M= \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right) }. \]Example: Midpoint in Space
Find the midpoint of
\[ (-2,4,7) \]and
\[ (6,-2,1). \]Average corresponding coordinates:
\[ M= \left( \frac{-2+6}{2}, \frac{4+(-2)}{2}, \frac{7+1}{2} \right). \]Therefore,
\[ \boxed{M=(2,1,4)}. \]Displacement Vectors
The displacement from
\[ P=(x_1,y_1,z_1) \]to
\[ Q=(x_2,y_2,z_2) \]is
\[ \boxed{ \overrightarrow{PQ} = \langle x_2-x_1,\, y_2-y_1,\, z_2-z_1 \rangle }. \]The distance \(PQ\) is the magnitude of this vector.
Vector Magnitude in Three Dimensions
For
\[ \mathbf{v}=\langle a,b,c\rangle, \]its magnitude is
\[ \boxed{ \|\mathbf{v}\| = \sqrt{a^2+b^2+c^2} }. \]This is the same Pythagorean structure as the distance formula.
Lines in Three Dimensions
A line in space can be described by a point
\[ \mathbf{r}_0= \langle x_0,y_0,z_0\rangle \]and a nonzero direction vector
\[ \mathbf{v}= \langle a,b,c\rangle. \]Its vector equation is
\[ \boxed{ \mathbf{r} = \mathbf{r}_0+t\mathbf{v} }. \]Parametric Equations of a Spatial Line
The vector equation becomes
\[ \boxed{x=x_0+at}, \] \[ \boxed{y=y_0+bt}, \] \[ \boxed{z=z_0+ct}. \]As \(t\) ranges over the real numbers, these equations generate every point on the line.
Example: Spatial Line
A line passes through
\[ (1,-2,4) \]with direction vector
\[ \langle3,5,-1\rangle. \]Its parametric equations are
\[ \boxed{ x=1+3t,\qquad y=-2+5t,\qquad z=4-t }. \]Planes in Three Dimensions
A plane is a flat two-dimensional surface extending infinitely through three-dimensional space.
A standard plane equation is
\[ \boxed{Ax+By+Cz=D}, \]where \(A\), \(B\), and \(C\) are not all zero.
Normal Vector of a Plane
For
\[ Ax+By+Cz=D, \]the vector
\[ \boxed{\mathbf{n}=\langle A,B,C\rangle} \]is perpendicular to the plane.
This vector is called the plane's normal vector.
Point-Normal Form of a Plane
If a plane passes through
\[ (x_0,y_0,z_0) \]and has normal vector
\[ \langle A,B,C\rangle, \]then
\[ \boxed{ A(x-x_0)+ B(y-y_0)+ C(z-z_0)=0 }. \]Example: Equation of a Plane
Find the plane through
\[ (2,1,-3) \]with normal vector
\[ \langle4,-2,5\rangle. \]Use point-normal form:
\[ 4(x-2)-2(y-1)+5(z+3)=0. \]Simplify:
\[ 4x-2y+5z+9=0. \]Therefore,
\[ \boxed{4x-2y+5z=-9}. \]Coordinate Planes as Special Plane Equations
The coordinate planes are special cases of
\[ Ax+By+Cz=D. \]Specifically,
\[ \boxed{xy\text{-plane}: z=0}, \] \[ \boxed{xz\text{-plane}: y=0}, \] \[ \boxed{yz\text{-plane}: x=0}. \]Planes Parallel to Coordinate Planes
The equation
\[ z=k \]describes a plane parallel to the \(xy\)-plane.
Likewise,
\[ y=k \]is parallel to the \(xz\)-plane, and
\[ x=k \]is parallel to the \(yz\)-plane.
Spheres
A sphere is the set of all points in three-dimensional space at a fixed distance \(r\) from a fixed center
\[ (h,k,\ell). \]The distance condition is
\[ \sqrt{ (x-h)^2+ (y-k)^2+ (z-\ell)^2 }=r. \]Standard Equation of a Sphere
Squaring gives
\[ \boxed{ (x-h)^2+ (y-k)^2+ (z-\ell)^2 = r^2 }. \]This is the direct three-dimensional analogue of the circle equation.
Example: Reading a Sphere Equation
Consider
\[ (x-2)^2+(y+3)^2+(z-5)^2=49. \]The center is
\[ \boxed{(2,-3,5)} \]and the radius is
\[ \boxed{7}. \]Sphere Centered at the Origin
If the center is the origin, the equation becomes
\[ \boxed{x^2+y^2+z^2=r^2}. \]Every point satisfying this equation lies exactly \(r\) units from \((0,0,0)\).
Cross-Sections of a Sphere
Intersecting a sphere with a coordinate plane produces a circle when the plane passes through or sufficiently near the sphere.
For
\[ x^2+y^2+z^2=25, \]setting
\[ z=0 \]gives
\[ x^2+y^2=25. \]The \(xy\)-cross-section is therefore a circle of radius \(5\).
Traces of Surfaces
A trace is the intersection of a surface with a selected plane.
Setting one coordinate equal to a constant can reduce a three-dimensional surface equation to a two-dimensional curve.
Traces are a powerful way to understand surfaces that are difficult to visualize all at once.
Example: Traces of a Paraboloid
Consider
\[ z=x^2+y^2. \]Setting
\[ y=0 \]gives
\[ z=x^2, \]a parabola in the \(xz\)-plane.
Setting
\[ z=c \]gives
\[ x^2+y^2=c, \]a circle when \(c>0\).
Cylinders in Three Dimensions
An equation involving only two of the three coordinates often describes a cylinder extending parallel to the missing coordinate axis.
For example,
\[ x^2+y^2=9 \]contains no \(z\).
Therefore it describes a circular cylinder of radius \(3\) extending parallel to the \(z\)-axis.
Why a Missing Variable Creates Extension
In
\[ x^2+y^2=9, \]the value of \(z\) is unrestricted.
Any point
\[ (x,y) \]satisfying the circle equation may be paired with every real value of
\[ z. \]The two-dimensional circle is therefore extended through space.
Distance to Coordinate Planes
For a point
\[ P=(x,y,z), \]the perpendicular distance to the \(xy\)-plane is
\[ \boxed{|z|}. \]Similarly, the distances to the other coordinate planes are
\[ \boxed{|y|} \]for the \(xz\)-plane and
\[ \boxed{|x|} \]for the \(yz\)-plane.
Distance to Coordinate Axes
The distance from
\[ (x,y,z) \]to the \(z\)-axis depends only on the \(x\)- and \(y\)-coordinates:
\[ \boxed{\sqrt{x^2+y^2}}. \]Analogously, the distance to the \(x\)-axis is
\[ \boxed{\sqrt{y^2+z^2}}, \]and the distance to the \(y\)-axis is
\[ \boxed{\sqrt{x^2+z^2}}. \]Equality of Ordered Triples
Two ordered triples are equal exactly when corresponding coordinates are equal:
\[ \boxed{ (a,b,c)=(d,e,f) \iff a=d,\ b=e,\ c=f }. \]For example, if
\[ (x,2y,z-1)=(4,10,6), \]then
\[ \boxed{x=4,\qquad y=5,\qquad z=7}. \]Spatial Coordinates and Data
Three coordinates can encode more than physical location.
A point
\[ (x,y,z) \]may represent three measured variables in a data set, three state variables in a model, or three components of a vector.
Coordinate geometry therefore provides a language that extends well beyond literal physical space.
From Three Dimensions to Higher Dimensions
The pattern continues.
A point in \(n\)-dimensional Cartesian space is represented by
\[ \boxed{ (x_1,x_2,\ldots,x_n) }. \]Its distance from the origin is
\[ \boxed{ \sqrt{x_1^2+x_2^2+\cdots+x_n^2} }. \]Visualization becomes difficult, but the algebraic structure remains consistent.
Cylindrical Coordinates Preview
Three-dimensional space can also be described using cylindrical coordinates
\[ \boxed{(r,\theta,z)}. \]The first two coordinates are polar coordinates in the \(xy\)-plane, while \(z\) records vertical position.
The conversions are
\[ x=r\cos\theta, \qquad y=r\sin\theta, \qquad z=z. \]Spherical Coordinates Preview
Spherical coordinates describe a point using one radial distance and two angles.
Different fields use different angle conventions, so notation must always be defined carefully.
The central idea is that three-dimensional location can be expressed in terms of distance from the origin and angular direction rather than three perpendicular distances.
Coordinate Systems Are Representations
A geometric point exists independently of the particular coordinates used to describe it.
Cartesian, cylindrical, and spherical coordinates can assign different numerical descriptions to the same location.
Choosing a coordinate system is therefore a modeling decision.
Common Error: Reordering the Coordinates
The points
\[ (2,3,5) \]and
\[ (5,3,2) \]are generally different.
The first, second, and third coordinates have assigned geometric roles and cannot be reordered arbitrarily.
Common Error: Confusing an Axis with a Plane
The equation
\[ z=0 \]describes the entire \(xy\)-plane, not the \(x\)-axis.
The \(x\)-axis requires both
\[ y=0 \]and
\[ z=0. \]Common Error: Omitting a Coordinate Difference in Distance
Three-dimensional distance requires all three coordinate changes:
\[ \sqrt{ (\Delta x)^2+ (\Delta y)^2+ (\Delta z)^2 }. \]Using only \(\Delta x\) and \(\Delta y\) computes the distance between projections onto the \(xy\)-plane, not the full spatial distance.
Common Error: Treating a Sphere as a Circle
The equation
\[ x^2+y^2+z^2=25 \]describes a sphere in \(\mathbb{R}^3\).
The equation
\[ x^2+y^2=25 \]describes a circle in \(\mathbb{R}^2\), but in \(\mathbb{R}^3\) it describes a cylinder because \(z\) is unrestricted.
Common Error: Assuming Octant Numbering Is Universal
The sign pattern of a point is always unambiguous.
Octant numbering beyond the first octant is not as standardized as quadrant numbering in the plane.
When precision matters, state the coordinate signs directly.
A Reliable Procedure for Reading a Point in Space
- Read the \(x\)-coordinate.
- Read the \(y\)-coordinate.
- Read the \(z\)-coordinate.
- Preserve the order \((x,y,z)\).
- Use coordinate signs to determine directional placement.
- Check whether any zero coordinate places the point on a coordinate plane or axis.
A Reliable Procedure for Spatial Distance
- Label the two points consistently.
- Compute \(x_2-x_1\).
- Compute \(y_2-y_1\).
- Compute \(z_2-z_1\).
- Square all three differences.
- Add the squares.
- Take the nonnegative square root.
- Simplify the radical if possible.
Practice 1: Coordinate Plane
Which coordinate plane contains
\[ (4,-7,0)? \]Practice 2: Coordinate Axis
Which coordinate axis contains
\[ (0,0,-6)? \]Practice 3: Projection
Project
\[ (3,-5,8) \]onto the \(xy\)-plane.
Practice 4: Distance from the Origin
Find the distance from
\[ (2,-3,6) \]to the origin.
Practice 5: Distance Between Two Points
Find the distance between
\[ A=(1,2,-1) \]and
\[ B=(4,6,7). \]Practice 6: Midpoint
Find the midpoint of
\[ (-2,5,1) \]and
\[ (8,-1,9). \]Practice 7: Sphere
Identify the center and radius of
\[ (x+3)^2+(y-2)^2+(z+1)^2=16. \]Practice 8: Plane
Write the equation of the plane through
\[ (1,0,2) \]with normal vector
\[ \langle2,-1,3\rangle. \]Practice 9: Spatial Line
Write parametric equations for the line through
\[ (-1,4,2) \]with direction vector
\[ \langle5,2,-3\rangle. \]Practice 10: Identify the Error
A student says that
\[ x^2+y^2=9 \]describes a sphere of radius \(3\) in three-dimensional space.
Explain the error.
Practice Answers
Practice 1
Since
\[ z=0, \]the point lies on the
\[ \boxed{xy\text{-plane}}. \]Practice 2
Since
\[ x=0,\qquad y=0, \]the point lies on the
\[ \boxed{z\text{-axis}}. \]Practice 3
Projection onto the \(xy\)-plane sets \(z=0\):
\[ \boxed{(3,-5,0)}. \]Practice 4
\[ d= \sqrt{2^2+(-3)^2+6^2} = \sqrt{4+9+36} = \boxed{7}. \]Practice 5
\[ d= \sqrt{ (4-1)^2+ (6-2)^2+ (7-(-1))^2 }. \] \[ d= \sqrt{3^2+4^2+8^2} = \sqrt{89}. \]Therefore,
\[ \boxed{\sqrt{89}}. \]Practice 6
\[ M= \left( \frac{-2+8}{2}, \frac{5+(-1)}{2}, \frac{1+9}{2} \right). \]Therefore,
\[ \boxed{M=(3,2,5)}. \]Practice 7
Compare with
\[ (x-h)^2+(y-k)^2+(z-\ell)^2=r^2. \]Therefore the center is
\[ \boxed{(-3,2,-1)} \]and the radius is
\[ \boxed{4}. \]Practice 8
Use point-normal form:
\[ 2(x-1)-y+3(z-2)=0. \]Simplify:
\[ 2x-y+3z-8=0. \]Therefore,
\[ \boxed{2x-y+3z=8}. \]Practice 9
\[ \boxed{ x=-1+5t,\qquad y=4+2t,\qquad z=2-3t }. \]Practice 10
The equation contains no \(z\)-term, so \(z\) is unrestricted.
Therefore
\[ x^2+y^2=9 \]describes a circular cylinder of radius \(3\) extending parallel to the \(z\)-axis.
A sphere of radius \(3\) centered at the origin would require
\[ \boxed{x^2+y^2+z^2=9}. \]What to Remember
Three-dimensional Cartesian coordinates use
\[ \boxed{(x,y,z)}. \]The coordinate planes are
\[ \boxed{z=0,\qquad y=0,\qquad x=0}. \]Distance between
\[ (x_1,y_1,z_1) \]and
\[ (x_2,y_2,z_2) \]is
\[ \boxed{ d= \sqrt{ (x_2-x_1)^2+ (y_2-y_1)^2+ (z_2-z_1)^2 } }. \]The midpoint is
\[ \boxed{ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2} \right) }. \]A sphere centered at \((h,k,\ell)\) has equation
\[ \boxed{ (x-h)^2+(y-k)^2+(z-\ell)^2=r^2 }. \]The key principle is that the familiar coordinate ideas from the plane extend coordinate by coordinate into space.
Where We Go Next
Coordinates let us represent points, lines, planes, spheres, cylinders, and more. The next step is to turn geometric conditions themselves into equations and inequalities.
The next lesson, Algebraic Descriptions of Geometric Loci, develops how distance, alignment, symmetry, boundaries, and other geometric constraints become algebraic descriptions of entire sets of points.