YouTube - Notes - Full video and playlist - Vectors and the Geometry of Space playlist
In this video I find the sum of an infinite series of consecutive magnitudes of vector projections and show it turns out to be just a geometric series. I first calculate the magnitudes for the first few vector projections and realize it follows a distinct pattern. This allows us to write the sum as a geometric series, thus we can just use the formula for the sum of a convergent geometric series. At the end of the video I also go over a quick recap and visual illustration of the geometric series.
Timestamps:
Problem 4: Sum of vector projections: 0:00
Solution: Recap on the dot and vector projections: 0:48
Calculate projections to find a pattern: 2:26
General equation for the magnitude of the n-th vector: 12:50
Infinite sum of the magnitude of the n-th vector: 15:09
Sum is a geometric series: 17:46
Sum is equal to 20: 18:59
Recap on the Geometric series: 19:43
Problems Plus 5: Infinite Sum of Vector Projections is a Geometric Series