Section 2.1 - Three-Dimensional Coordinate Systems

Section 2.1 - Three-Dimensional Coordinate Systems

3D Space

In analytic geometry, three-dimensional space is described using three mutually perpendicular axes: the x-axis, y-axis, and z-axis. These axes intersect at the origin, denoted as O, and define the coordinate system for locating points in space.

• Coordinate Axes: The x, y, and z axes are perpendicular to each other and intersect at the origin.

• Coordinate Planes: The xy-plane contains the x- and y-axes; the yz-plane contains the y- and z-axes; the xz-plane contains the x- and z-axes.

• Octants: The three coordinate planes divide space into eight regions called octants.

• Point Representation: Any point P in space is represented by an ordered triple $(x, y, z)$, where x, y, and z are the coordinates along the respective axes.

Example: The point P(3, -2, 5) is located 3 units along the x-axis, -2 units along the y-axis, and 5 units along the z-axis from the origin.

Distance Formula in 3D

The distance between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in three-dimensional space is given by:

$|P_1P_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

• Application: Used to find the straight-line distance between any two points in 3D space.

Example: Find the distance from $P_1(-2, 5, 0)$ to $Q(1, -1, 7)$.

Equations of Planes Parallel to Coordinate Planes

Planes in 3D space can be parallel to one of the coordinate planes. Their equations are simple forms:

• Parallel to xy-plane: The plane through $(a, b, c)$ parallel to the xy-plane is $z = c$.

• Parallel to xz-plane: The plane through $(a, b, c)$ parallel to the xz-plane is $y = b$.

• Parallel to yz-plane: The plane through $(a, b, c)$ parallel to the yz-plane is $x = a$.

Example: Write the equation of the plane passing through $(3, 11, 7)$ that is parallel to the yz-plane. The equation is $x = 3$.

Equation of a Sphere

A sphere in 3D space is defined as the set of all points that are a fixed distance (radius) from a given center $(h, k, l)$. The equation of a sphere is:

$(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2$

• Center: $(h, k, l)$

• Radius: $r$

• Special Case: If the center is the origin $(0, 0, 0)$, the equation simplifies to $x^2 + y^2 + z^2 = r^2$.

Example: Find the equation of a sphere with radius 2 and center $(2, -1, 5)$: $(x - 2)^2 + (y + 1)^2 + (z - 5)^2 = 4$

Example: Show that $x^2 + y^2 + z^2 + 4x - 6y + 2z + 6 = 0$ is the equation of a sphere, and find its center and radius.

Summary Table: Equations of Planes Parallel to Coordinate Planes

Plane Orientation	Equation	Parallel to
xy-plane	$z = c$	xy-plane
xz-plane	$y = b$	xz-plane
yz-plane	$x = a$	yz-plane

Additional info:

• In 3D analytic geometry, equations of planes and spheres are foundational for understanding surfaces and volumes.

• These concepts are essential for later topics such as vectors, lines, and calculus in multiple dimensions.