Vectors and the Geometry of Space: The Cross Product

in Threespeak2 months ago

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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:!As32ynv0LoaIiahlV-ELBa7AO4z_vw?e=r63WU4

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Related Videos:

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▶️ 3Speak


In this video I go further into Vectors and Geometry of Space and this time do an in-depth examination of the Cross Product. This is also one of the most useful concepts in physics so make sure to fully grasp it!

View video notes on the Hive blockchain:

View video sections playlist:

View the full Vectors and Geometry of Space video series: