Calculus 3 — 3.1: The Three-Dimensional Coordinate System

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This lesson builds the three-dimensional coordinate system — R³ — from the ground up. Starting from the familiar XY plane, a third perpendicular axis (Z) is introduced, the right-hand rule fixes the system's orientation, and two step-by-step methods show how to plot any point in space. By the end, you can locate any ordered triple in 3D and understand the geometry behind the construction.

Key concepts covered:
• Extending the 2D XY plane into three-space by adding a mutually perpendicular Z axis
• Standard drawing convention: positive X toward the viewer, positive Y to the right, positive Z upward
• The right-hand rule — curl positive X toward positive Y and the thumb points along positive Z
• Why swapping two axes produces a left-handed system and reverses orientation
• The three coordinate planes: XY (horizontal), YZ (front vertical), and XZ (side vertical)
• Eight octants in 3D versus four quadrants in 2D, and why the count doubles
• Ordered triples (x, y, z) as the three-coordinate address for every point in space
• Plotting method 1: locate the projection on the XY plane, then rise by the Z coordinate
• Plotting method 2: draw the segment from the origin to the projection, then translate it upward
• Why dashed construction lines are essential for conveying depth in a flat sketch

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SOURCE MATERIALS
The source materials for this video are from https://www.youtube.com/watch?v=ZAv3bF2GznI

Видео Calculus 3 — 3.1: The Three-Dimensional Coordinate System канала Ludium