Vectors and the Geometry of Space: The Cross Product

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

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