Name: Vectors and the Geometry of Space: The Cross Product
Uploaded: 2023-01-31T04:57:36+00:00
Duration: 2 h 50 min 53 s
Description: 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 t

1 year 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