In this video I go over scalar multiplication for vectors. The idea behind scalar multiplication is wanting to multiply a vector by a real number c such that it becomes shorter or longer but in the same parallel direction. In other words, the real number c "scales" the vector. And when multiplying by a negative number, the vector flips in the opposite direction. When dealing with negative vectors, we can visualize this by subtracting vectors using the Parallelogram or Triangle Laws. I also go over an example to illustrate scaling and adding vectors.
The timestamps of key parts of the video are listed below:
- Definition of Scalar Multiplication: 0:00 - Negative Vectors: 3:00 - Example 2: 6:11
This video was taken from my earlier video listed below:
In this video I go over a list of 8 properties of vectors in terms of addition and scalar multiplication. I also provide algebraic proofs for properties 1 and 2. The remaining properties, however, are fairly straightforward and so I only briefly go over them. Property 1, a + b = b + a, be seen from the Parallelogram Law and Property 2 (called the associative law), a + (b + c) = (a + b) + c, can be proved by repeated use of the Triangle Law.
The timestamps of key parts of the video are listed below:
- List of 8 Vector Properties: 0:00
- Proof of Property 1: 1:39
- Proof of Property 2: 4:32
- Other Properties: 8:55
This video was taken from my earlier video listed below:
- Vectors and the Geometry of Space: Vectors: https://youtu.be/0lJiVlKw4Kc
- Video notes: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-vectors
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EiRECLVkBCcFhSpbSz8XIw
Related videos:
Vectors and Geometry of Space video series: https://www.youtube.com/playlist?list=PLai3U8-WIK0FjJpwnxwdrOR7L8Ul8VZoZ .
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https://www.youtube.com/watch?v=fvzgdGsdIXM
In this video I go over an infinite series that involves natural logarithms and can be solved by rearranging it into a form that allows for a Telescoping Sum to be applied. Recall that the Telescoping Sum is such that an infinite series has all terms cancel out except the first and the last term. When we rearrange, using log rules, the series, we see that it can be rewritten into terms whose individual terms are separate by an integer; which means they will cancel out with each successive series addition. Truly remarkable stuff!
The timestamps of key parts of the video are listed below:
- Problem 11: 0:00
- Solution: Simplifying using Logarithm Rules 0:18
- Telescoping Sum: 6:11
- Sum is - ln 2: 10:29
This video was taken from my earlier video listed below:
- Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P
Related Videos:
Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .
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In this video I go over another example on parabolas, and this time go over a very useful application of parabolas, which is the property that lines extending from the focus of a parabola to a point on the parabola get protruded in a horizontal line. This property makes it very useful to design headlights and telescopes in a paraboloid shape; which is a 3D parabola formed by rotating a parabola about it’s central axis. In this particular video I prove that this is the case by proving the angle between a line tangent to point on the parabola and a line extending to that point from the focus is equivalent to the angle between the tangent line and a horizontal line protruding from that point on the parabola. Many cars and large-scale mirror telescopes use this design property of the parabola, and thus it is pretty cool to see the math behind just why it works! This is a very interesting video in it is a more advanced example, part of the Problems Plus section of my Calculus book, as well as it’s real world applications so make sure to watch this video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIhvh4UkkM5vzwV2H3Yw
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/video-notes-conic-sections-parabolas-example-2-paraboloid-headlights-and-telescopes-problems-plus
Related Videos:
Conic Sections: Parabolas: Example 1: y^2 + 10x = 0: https://youtu.be/JnSB2aVzxoE
Conic Sections: Parabolas: Definition and Formula: https://youtu.be/kCJjXuuIqbE
Angle Between Two Lines: Formula in terms of Slopes: https://youtu.be/O8M4ZErxE-M .
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https://www.youtube.com/watch?v=Frd94l5Lfjs
In this video I go over determinants, what they are, and then show that we can write the Cross Product in a simpler and easier to remember form as a determinant. I also go over 2 examples determining the cross product using determinants.
The timestamps of key parts of the video are listed below:
- Definition of Determinants: 0:00
- Cross Product in Determinant Form: 6:46
- Example 1: 12:25
- Example 2: 15:36
This video was taken from my earlier video listed below:
- Vectors and the Geometry of Space: The Cross Product: https://youtu.be/k8GRt95i-Gc
- Video notes: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-cross-product
- Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FuRJ8rg-YVQvfPoPOwhuRW .
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https://www.youtube.com/watch?v=rdityjLg7bI
In this video I go over some examples on finding derivatives of logarithmic functions, especially natural logs.
Download the notes in my video: https://www.dropbox.com/s/931zulq7kltl878/428%20-%20Derivatives%20of%20Logs%20Examples.pdf
Related Videos:
Inverse Functions - f-1(x) - An Introduction: http://youtu.be/qIqj3oKwFi8
Logarithms and their Properties - An Introduction: http://youtu.be/AZ6KKym19gI
Natural Logarithms, Log base 10, and Some Examples Using Logs: http://youtu.be/XRSkMk5L3pk
Derivative of y = Log(x) and y = Ln(x): http://youtu.be/5e6MisvvMPE
Logarithmic Differentiation: Easy way to simplify derivatives: http://youtu.be/_hfZdRztUjU .
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https://www.youtube.com/watch?v=bAU_BhSEpnE
In this video I promote my Telegram channel MES Links and its associated Discussion group. Recently I had created a Telegram channel and found it to be very useful to quickly share lots of links, images, and videos in one place and in real-time while I do my typical extensive research. This is a great way to keep track with what I am up to and currently researching. I post lots of information relating to mathematics, science, memes, and conspiracy topics so there is something for everyone!
The links to the channel and group are shown below:
MES Links channel: https://t.me/meslinks
MES Links Discussion group: https://t.me/meslinkschat
Note that the group allows users to post and discuss with other users, as well as comment on posts from the channel.
Also included in this video is a quick tutorial on managing the Telegram storage settings on the Telegram desktop and Android app. The Telegram apps can be downloaded through their official website: https://telegram.org/apps
The topics covered as well as their timestamps are listed below.
- Introduction: 0:00
1. MES Links Telegram Channel and Discussion Group: 0:31
2. Telegram Storage Settings: 4:58
Related Videos:
MES Truth YouTube channel: https://youtube.com/@mestruth
Alternative Links to Avoid YouTube Censorship: https://peakd.com/hive-106474/@mestruth/alternative-platforms-to-avoid-youtube-censorship .
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In this video I go over the law of sines again but this time write it in the more advanced and less common form which includes the diameter of the corresponding circumcircle that is formed around any triangle. This version of the law of sines states:
a/sinA = b/sinB = c/sinC = D
where:
- a, b, c are the lengths of the triangle
- A, B, C are the corresponding opposite angles
- D is the diameter of the circumcircle
In this proof, I utilize the basic proof of the law of sines as well as taking into account the inscribed angle theorem which allows us to include the Diameter of the circumcircle in the relation between the angles and lengths of the triangle.
Download the notes in my video: http://1drv.ms/1BAvWLC
View Video Notes on Steemit: https://steemit.com/mathematics/@mes/law-of-sines-circumcircle-proof
Related Videos:
Law of Sines: Basic Proof: http://youtu.be/_HDM60RRh8U
Inscribed Angle Theorem: Corollary Properties: http://youtu.be/ONHVPKkdGGI
Inscribed Angle Theorem: Inscribed on Minor Arc: http://youtu.be/53ueXQa8OEM
Inscribed Angle Theorem: http://youtu.be/XZKogfeUSlY .
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...
https://www.youtube.com/watch?v=Xeo0A_xDCeQ
In this video I go over another example on implicit differentiation and now look at the famous cubic function, the Folium of Descartes. I also go over some very interesting historical facts in the history of math so make sure you watch this video!
Download the notes in my video: https://www.dropbox.com/s/c5e55yaf9fsw7gd/417%20-%20Implicit%20Differentiation%20Example%20Descartes.pdf
Related Videos:
Implicit Differentiation Example on Equation of a Circle: http://youtu.be/4_YzO6IsloY
Implicit Differentiation - A Brief Introduction: http://youtu.be/12OY1b3DYHQ
Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU
Derivative of y = x^n: Power Rule Part 1: n is positive integer: http://youtu.be/-Yv85MZNYgU
Power Functions and their Properties Part 1 - A Simple Explanation: http://youtu.be/2MKko4ZkSf0
Limit Laws - Part 2 + Brief History of Newton and Limits: http://youtu.be/K5n7y2BvnOs .
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...
https://www.youtube.com/watch?v=D3ahNyg-OoQ
In this video I go over the famous Book Stacking Problem, and show that it is possible to stack books that extend infinitely far away from the table! To do this you have to stack such that the first book extends 1/2 from the second book which extends 1/4 from the third which extends 1/6 from the 4th, and so one. Then the center of mass of the stack of books can be determined and shown to be still above the table, which means the books won't tip over. Lastly, I show that the series of distances 1/2 + 1/4 + 1/6 + ... etc. is a divergent Harmonic series, which means that the distance keeps extending out to infinity as we add more books. Try this yourself and let me know how many books, or playing cards, you can stack up!
The timestamps of key parts of the video are listed below:
- Problem 12: Book Stacking Problem: 0:00
- Solution: Book extending completely off the table: 1:57
- Center of mass of the stack of books: 7:20
- Center of mass doesn't extend past the table: 17:02
- Books can extend infinitely far via the divergent Harmonic series: 18:42
This video was taken from my earlier video listed below:
- Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P
Related Videos:
Sequences and Series playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .
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...
https://www.youtube.com/watch?v=_2IAh41R1sg