Vectors and the Geometry of Space: The Cross Product

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    mes

    Published on Jan 30, 2023
    About :

    In this video I go over further into Vectors and the Geometry of Space and this time cover the Cross Product. The cross product is a very useful concept and is defined such that it produces a vector that is perpendicular to 2 other vectors. This property is very useful in physics, especially when dealing with rotation and torque.

    While the definition of the cross product involves a lot of algebra, I show that it can be made easier to both remember and calculate by writing it in determinant form. Furthermore, I show that the length of the cross product is equal to the parallelogram formed from 2 vectors. I also go over several other theorems and topics that follow from the cross product, such as the right-hand rule, the scalar triple product, and a number of properties of the cross product. This is a great in-depth introduction and examination of the cross product so make sure to watch this whole video!

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

    • Introduction: 0:00
    • Calculus Book Reference: 0:44
    • Sections in Calculus Book Chapter: 1:06
    • Topics to Cover: 1:38
    1. The Cross Product: 3:01
      • Definition: 15:36
      • Historical Note: Sir William Rowan Hamilton: 16:55
    2. Determinants: 18:01
      • Cross Product in Determinant Form: 24:48
      • Example 1: 30:27
      • Example 2: 33:38
    3. Cross Product Orthogonality Proof: 36:50
      • Theorem 1: 37:19
    4. Right-Hand Rule: 43:07
    5. Cross Product Length: 44:58
      • Theorem 2: 45:23
    6. Corollary: 1:11:47
    7. Geometric Interpretation of Theorem 2: 1:15:19
      • Example 3: 1:19:46
      • Example 4: 1:31:44
    8. Properties of the Cross Product: 1:35:03
      • Theorem 3: 1:45:36
      • Proof of Property 5: 1:47:33
    9. Triple Products: 1:53:09
      • Formula of a Parallelepiped: 2:01:08
      • Example 5: 2:02:05
    10. Torque: 2:07:00
      • Example 6: 2:10:31
    11. Exercises: 2:14:42
      • Exercise 1: 2:15:20
      • Exercise 2: 2:18:23
      • Exercise 3: 2:22:12
      • Exercise 4: 2:28:16
      • Exercise 5: Vector Triple Product: 2:34:08

    Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiahlV-ELBa7AO4z_vw?e=r63WU4

    View video notes on the Hive blockchain: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-cross-product

    View video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FuRJ8rg-YVQvfPoPOwhuRW

    Related Videos:

    Vectors and the Geometry of Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ

    Vectors and the Geometry of Space: 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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