Discovery Project: The Geometry of a Tetrahedron

In this video I go over the Discovery Project titled The Geometry of a Tetrahedron. Discovery Projects are advanced math problems in each chapter of my calculus book, and this one is part of the Vectors and Geometry of Space chapter. This project involves 3 Questions regarding the tetrahedron, which is a triangular pyramid. The first question involves finding the areas of each triangular face using the Cross Product. The second question involves finding the volume of a tetrahedron when written in coordinate vector form. And the third question involves deriving the 3D version of the Pythagorean Theorem, which is: the 3 triangular areas squared at a vertex sum up to equal the square of the opposite triangular area!

The topics covered as well as their timestamps are listed below.

• Introduction: 0:00
• Calculus Book Reference: 0:56
• Sections in Calculus Book Chapter: 1:12
• Topics to Cover: 2:03

1. Discovery Project: The Geometry of a Tetrahedron: 2:25
2. 3 Questions: 3:05
   • Question 1: 5:00
   • Question 2: 34:59
     • Solution to Part (a): 36:10
       • The Equation of a Plane: 41:20
       • The Distance from a Point to a Plane: 45:31
     • Solution to Part (b): 52:43
   • Question 3: 1:05:35

Download video notes: https://1drv.ms/b/s!As32ynv0LoaIicNsAiLSiBI1ivS39w?e=V33CkF

View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/discovery-project-geometry-of-a-tetrahedron

View video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GoEi9wxl8nTFcfw1ay-_1T

View the full Vectors and the Geometry of Space series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ

Related Videos:

The Cross Product: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-cross-product

The Dot Product: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-dot-product

Complex Numbers as Rotation Matrices: https://peakd.com/hive-128780/@mes/complex-numbers-as-rotation-matrices

Vectors: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-vectors

3D Coordinate Systems: https://peakd.com/hive-128780/@mes/vector-space-and-geometry-3d-coordinate-systems .

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