Vectors and the Geometry of Space: The Dot Product

Math Easy Solutions

description

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

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
Vector Angles: 56:49
- Direction Angles and Direction Cosines: 1:02:55
- Example 5: 1:17:28
Projections: Vector and Scalar Projections: 1:21:27
- Summary of Projections: 1:30:22
- Example 6: 1:33:19
Vectors in Physics: 1:39:04
- Example 7: 1:42:33
- Example 8: 1:45:25