Introduction to Three-Dimensional Geometry

Cartesian Coordinate System in Three Dimensions (Axes, Planes, Octants)

Expanding upon the concept of locating points in a two-dimensional plane using a pair of perpendicular axes, the Cartesian Coordinate System in Three Dimensions provides a framework for specifying the position of any point in space. This system uses three mutually perpendicular axes that intersect at a single origin.

Coordinate Axes

• The system consists of three mutually perpendicular straight lines. These lines serve as the reference axes for measuring distances in three independent directions.

• These three lines are called the coordinate axes. By convention, they are labeled as the x-axis, the y-axis, and the z-axis.

• The unique point where all three axes intersect is called the origin of the coordinate system. It is the reference point from which all distances are measured and is denoted by $O$. In three dimensions, the coordinates of the origin are $(0, 0, 0)$.

• The positive and negative directions along each axis are typically indicated by arrowheads. A standard convention for orienting the positive directions of the axes in 3D space is the right-hand rule. If you align your right hand so that your fingers curl from the positive x-axis towards the positive y-axis, your extended thumb will point in the direction of the positive z-axis. This defines a right-handed coordinate system, which is the most commonly used.

Coordinate Planes

The three coordinate axes, when taken two at a time, define three mutually perpendicular planes. These planes are called the coordinate planes, and they effectively divide the 3D space into different regions.

• XY-plane: This plane contains the x-axis and the y-axis. Any point lying in the XY-plane has its z-coordinate equal to zero. The equation of the XY-plane is $\mathbf{z = 0}$. It is the "floor" or base plane in many visualizations.

• YZ-plane: This plane contains the y-axis and the z-axis. Any point lying in the YZ-plane has its x-coordinate equal to zero. The equation of the YZ-plane is $\mathbf{x = 0}$.

• XZ-plane: This plane contains the x-axis and the z-axis. Any point lying in the XZ-plane has its y-coordinate equal to zero. The equation of the XZ-plane is $\mathbf{y = 0}$.

These three planes intersect pairwise along the coordinate axes (XY and XZ intersect along the x-axis, XY and YZ intersect along the y-axis, and XZ and YZ intersect along the z-axis). All three planes intersect at the origin $O(0, 0, 0)$.

Octants

The three coordinate planes ($z=0$, $x=0$, and $y=0$) divide the entire three-dimensional space into eight distinct regions. These regions are analogous to the quadrants in a 2D system and are called octants (from the Latin word "octo" meaning eight).

Each octant is uniquely defined by the combination of the signs (+ or –) of the x, y, and z coordinates of the points that lie within that region. The origin and the points lying on the coordinate axes and planes are not considered to be within any octant.

The signs of the coordinates in each octant, following a common numbering convention (although other conventions exist), are as follows:

The first octant (Octant I) is the region where all three coordinates ($x, y, z$) are positive ($x>0, y>0, z>0$). The other octants are numbered based on the signs of the x and y coordinates relative to the first quadrant in the XY-plane (when viewed from above, i.e., from the positive z-axis), extended into the regions where z is positive (Octants I to IV) and where z is negative (Octants V to VIII).

Coordinates of a Point in Space

In the three-dimensional Cartesian coordinate system, the position of any point in space is uniquely determined by an ordered triplet of real numbers. These numbers are the point's coordinates and represent its directed distances from the three coordinate planes.

For a point $P$ in space, its coordinates are written as $(x, y, z)$.

• The x-coordinate ($x$) of point $P$ is the signed perpendicular distance of $P$ from the YZ-plane (the plane where $x=0$). It is positive if $P$ is on the positive side of the YZ-plane (in the direction of the positive x-axis) and negative if $P$ is on the negative side.
• The y-coordinate ($y$) of point $P$ is the signed perpendicular distance of $P$ from the XZ-plane (the plane where $y=0$). It is positive if $P$ is on the positive side of the XZ-plane (in the direction of the positive y-axis) and negative if $P$ is on the negative side.
• The z-coordinate ($z$) of point $P$ is the signed perpendicular distance of $P$ from the XY-plane (the plane where $z=0$). It is positive if $P$ is on the positive side of the XY-plane (in the direction of the positive z-axis) and negative if $P$ is on the negative side.

To visualize the location of a point $P(x, y, z)$:

Start at the origin $O(0, 0, 0)$.

Move $x$ units along the x-axis (positive direction if $x>0$, negative if $x<0$). Let's call this point $P_x(x, 0, 0)$.

From $P_x$, move $y$ units parallel to the y-axis (positive direction if $y>0$, negative if $y<0$). This takes you to a point in the XY-plane, $P_{xy}(x, y, 0)$.

From $P_{xy}$, move $z$ units parallel to the z-axis (positive direction if $z>0$, negative if $z<0$). The endpoint is the point $P(x, y, z)$.

Alternatively, you can imagine forming a rectangular box with one vertex at the origin and the opposite vertex at $P(x, y, z)$, such that the edges of the box are parallel to the coordinate axes. The lengths of the edges along the x, y, and z directions correspond to $|x|, |y|, |z|$.

Coordinates of Points on Axes and Planes

Points that lie on the coordinate axes or in the coordinate planes have one or two of their coordinates equal to zero:

• Points on the x-axis: Any point on the x-axis lies in both the XZ-plane (where $y=0$) and the XY-plane (where $z=0$). Therefore, its y-coordinate and z-coordinate must both be zero. Points on the x-axis have coordinates of the form $\mathbf{(x, 0, 0)}$.

• Points on the y-axis: Any point on the y-axis lies in both the YZ-plane (where $x=0$) and the XY-plane (where $z=0$). Therefore, its x-coordinate and z-coordinate must both be zero. Points on the y-axis have coordinates of the form $\mathbf{(0, y, 0)}$.

• Points on the z-axis: Any point on the z-axis lies in both the YZ-plane (where $x=0$) and the XZ-plane (where $y=0$). Therefore, its x-coordinate and y-coordinate must both be zero. Points on the z-axis have coordinates of the form $\mathbf{(0, 0, z)}$.

• Points on the XY-plane: Any point in the XY-plane is at a distance of 0 from the XY-plane. Therefore, its z-coordinate is zero. Points on the XY-plane have coordinates of the form $\mathbf{(x, y, 0)}$.

• Points on the YZ-plane: Any point in the YZ-plane is at a distance of 0 from the YZ-plane. Therefore, its x-coordinate is zero. Points on the YZ-plane have coordinates of the form $\mathbf{(0, y, z)}$.

• Points on the XZ-plane: Any point in the XZ-plane is at a distance of 0 from the XZ-plane. Therefore, its y-coordinate is zero. Points on the XZ-plane have coordinates of the form $\mathbf{(x, 0, z)}$.

• The Origin: The origin is the intersection of all three axes and all three coordinate planes. It has coordinates $\mathbf{(0, 0, 0)}$.

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Distance from Coordinate Planes and Axes

In the 3D Cartesian system, the coordinates of a point $P(x, y, z)$ not only specify its unique location but also directly provide information about its distances from the coordinate planes and axes. The distances considered here are the perpendicular distances, as the shortest distance from a point to a plane or a line is along the perpendicular segment.

Distance from Coordinate Planes

The coordinate planes are the YZ-plane ($x=0$), the XZ-plane ($y=0$), and the XY-plane ($z=0$). The coordinates of a point $P(x, y, z)$ are defined as its signed perpendicular distances from these planes. The actual (unsigned) distance is the absolute value of the corresponding coordinate.

For a point $P(x, y, z)$:

• The perpendicular distance from the YZ-plane ($x=0$) is the absolute value of its x-coordinate: $\mathbf{|x|}$.
• The perpendicular distance from the XZ-plane ($y=0$) is the absolute value of its y-coordinate: $\mathbf{|y|}$.
• The perpendicular distance from the XY-plane ($z=0$) is the absolute value of its z-coordinate: $\mathbf{|z|}$.

Distance from Coordinate Axes

The distance of a point $P(x, y, z)$ from a coordinate axis (x-axis, y-axis, or z-axis) is the length of the perpendicular segment from the point to that axis. The foot of this perpendicular on the axis is the point on the axis closest to $P$.

• Distance from the x-axis: The x-axis is defined by $y=0$ and $z=0$. The point on the x-axis closest to $P(x, y, z)$ is the point that shares the same x-coordinate as P, but has y and z coordinates equal to zero. This point is $A(x, 0, 0)$. The distance from $P(x, y, z)$ to the x-axis is the distance between $P(x, y, z)$ and $A(x, 0, 0)$. Using the distance formula in 3D:

  Distance from x-axis $= \sqrt{(x-x)^2 + (y-0)^2 + (z-0)^2} = \sqrt{0^2 + y^2 + z^2}$

  Distance from x-axis = $\mathbf{\sqrt{y^2 + z^2}}$

  

• Distance from the y-axis: The y-axis is defined by $x=0$ and $z=0$. The point on the y-axis closest to $P(x, y, z)$ is the point with the same y-coordinate as P, and x and z coordinates equal to zero. This point is $B(0, y, 0)$. The distance from $P(x, y, z)$ to the y-axis is the distance between $P(x, y, z)$ and $B(0, y, 0)$. Using the distance formula in 3D:

  Distance from y-axis $= \sqrt{(x-0)^2 + (y-y)^2 + (z-0)^2} = \sqrt{x^2 + 0^2 + z^2}$

  Distance from y-axis = $\mathbf{\sqrt{x^2 + z^2}}$

• Distance from the z-axis: The z-axis is defined by $x=0$ and $y=0$. The point on the z-axis closest to $P(x, y, z)$ is the point with the same z-coordinate as P, and x and y coordinates equal to zero. This point is $C(0, 0, z)$. The distance from $P(x, y, z)$ to the z-axis is the distance between $P(x, y, z)$ and $C(0, 0, z)$. Using the distance formula in 3D:

  Distance from z-axis $= \sqrt{(x-0)^2 + (y-0)^2 + (z-z)^2} = \sqrt{x^2 + y^2 + 0^2}$

  Distance from z-axis = $\mathbf{\sqrt{x^2 + y^2}}$

Summary Table: Distances for Point P(x, y, z)

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