Vectors and the Geometry of Space: The Dot Product

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    mes

    Published on Dec 18, 2022
    About :

    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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