1: Three-dimensional coordinate systems

Three-dimensional Coordinate Systems

In three-dimensional space, we extend the familiar 2D Cartesian plane by adding a third axis, enabling precise location of points in physical space. The Cartesian coordinate system uses three mutually perpendicular axes:

• x-axis (horizontal),
• y-axis (horizontal, perpendicular to x),
• z-axis (vertical, perpendicular to both).
  These axes intersect at the origin $(0, 0, 0)$, dividing space into eight octants. The first octant (where $x \geq 0$, $y \geq 0$, $z \geq 0$) is analogous to the first quadrant in 2D.

Key Principles

1. Right-Hand Rule:
   Point your right index finger along the positive x-axis and middle finger along positive y-axis. Your thumb then points toward positive z-axis. This convention standardizes orientation.

   • Violating this rule flips signs in cross products and rotation calculations.

2. Coordinates of a Point:
   A point $P$ is defined by $(x, y, z)$, where:

   • $x$ = signed distance from yz-plane,
   • $y$ = signed distance from xz-plane,
   • $z$ = signed distance from xy-plane.

3. Distance Formula:
   The distance $d$ between points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is:

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

   Derived from the Pythagorean theorem applied twice: first in the xy-plane, then extending vertically.

4. Midpoint Formula:
   The midpoint $M$ of segment $P_1P_2$ has coordinates:

   $$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)$$

Alternative Coordinate Systems

While Cartesian coordinates are foundational, other systems simplify specific problems:

• Cylindrical Coordinates: Use radius $r$, angle $\theta$, and height $z$ (ideal for cylinders/symmetry about z-axis).
• Spherical Coordinates: Use radius $\rho$, angles $\theta$ (longitude) and $\phi$ (latitude) (suited for spheres/radial symmetry).
  These will be detailed in Topic 5.

Applications

• Graphing Surfaces: Equations like $x^2 + y^2 + z^2 = r^2$ define spheres.
• Vector Operations: Coordinates enable vector addition, scalar multiplication, and magnitude calculations.
• Geometry: Planes and lines are described using coordinate equations (covered in Topic 3).

Exam Focus: Expect problems requiring distance calculations, midpoint determination, coordinate conversions, and visualization of surfaces. Mastery of the right-hand rule is essential for vector calculus.