Vectors and the Geometry of Space: The Dot Product

In this video I go over further into vectors and this time look at multiplying vectors with the dot product. In my previous videos I discussed adding vectors and multiplying vectors with a scalar. This time I discuss multiplication of two vectors via the dot product, which is defined as multiplying all corresponding vector components and adding them up, thus giving a scalar or real number value; aka not a vector. Further in the video I show that this way of defining vector multiplication has useful applications since it can be reinterpreted in terms of the angle between 2 vectors. I also discuss vector projections, direction cosines, and applications to physics of the dot product.

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

- Introduction: 0:00
- Sections in Calculus Book Chapter: 1:03
- Topics to Cover: 1:46
1. The Dot Product: 2:52
- Definition 1: Dot Product: 3:38
- Example 1: 6:15
- Properties of the Dot Product: 10:49
- Proof of Property 1: 12:42
- Proof of Property 2: 14:00
- Proof of Property 3: 15:59
- Proof of Property 4: 19:15
- Proof of Property 5: 24:00
- Geometric Interpretation of the Dot Product: 24:45
- Definition 2: Dot Product: 28:02
- Example 2: 43:14
- Corollary: 45:46
- Example 3: 46:49
- Perpendicular Vectors: 52:42
- Example 4: 54:47
2. Vector Angles: 56:49
- Direction Angles and Direction Cosines: 1:02:55
- Example 5: 1:17:28
3. Projections: Vector and Scalar Projections: 1:21:27
- Summary of Projections: 1:30:22
- Example 6: 1:33:19
4. Vectors in Physics: 1:39:04
- Example 7: 1:42:33
- Example 8: 1:45:25

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIiahlV-ELBa7AO4z_vw?e=7T9PN5

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

View video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0GbHgvpTY1mx61nTNO-Msl8

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

Related Videos:

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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