Relating dot product divided with square of the vector while changing basis of vector

The lecture (Mathematics for Machine Learning: Linear Algebra by Imperial College London ) covers changing basis of vectors. Finding difficult to follow how dot product is used while changing basis.

https://www.canva.com/design/DAE6AalLfdc/share/preview?token=WN6szDjbwy3-bLQ-fEJb_g&role=EDITOR&utm_content=DAE6AalLfdc&utm_campaign=designshare&utm_medium=link&utm_source=sharebutton<br />
Why dot product is divided by square of the length of vector to come to the conclusion that vector projection is 2 times b. I think first solving the below problems will help in the process.

https://www.canva.com/design/DAE6AYbwPQM/ExUqdS24TcITEgcwrPP4zw/view?utm_content=DAE6AYbwPQM&utm_campaign=designshare&utm_medium=link&utm_source=sharebutton<br />

Video link of lecture: https://d3c33hcgiwev3.cloudfront.net/3pBFJJOuEemGqAqLppoP9A.processed/full/360p/index.mp4?Expires=1646438400&Signature=YJfgW67qaeBZpuuirZEUbA53pr4~k3LzhCgxBpaIoEH-7qAvjzJ82kZedGp9cuWcwyS4Tb4hz9LDqAXNQGd9KhsUhounEwcsx4XEvakBiICoFeO15NN~aa8nrCfIIy58VT29V0otxg-TP8RJ7bKjWQ2lCiKmcuRKDcK8yvPM7rI_&Key-Pair-Id=APKAJLTNE6QMUY6HBC5A

Linear Algebra
Digitalsplendid Digitalsplendid
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  • Alessandro Iraci Alessandro Iraci
    0

    I'm fairly sure that the question is very easy, but I find it unlikely that anyone is going through the video for the current bounty. I think you should either increase it, or try to write your question clearly without referencing external sources.

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Erdos Erdos
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  • Erdos Erdos
    0

    Your post was kind of vague but I decided to answer as I thought this might be the first time you are asking a question here. Please make sure to be much more clear about your questions in future, otherwise you may not get a response.

  • Digitalsplendid Digitalsplendid
    0

    Thanks for the helpful answer. Is it true that the distance marked P in the diagram equal in magnitude? Also, it will help if you are kind enough to confirm the length of P starts from where a1 ends. If that be so, then I find some meaning breaking OM into Oa1 and a1M. That is, a1M = MN = P.

  • Digitalsplendid Digitalsplendid
    0

    Root 5 I understand is length of Oa1 and not ON.

  • Digitalsplendid Digitalsplendid
    0

    From another thread, the magnitude of MN appears 1 (https://mathforums.com/threads/relating-dot-product-divided-with-square-of-the-vector-while-changing-basis-of-vector.362851/#post-665818). An insight will help why I am confused.

  • Erdos Erdos
    0

    Regarding your first comment: Yes, the distance marked P in the diagram equal in magnitude of the Vector MN. Yes, I confirm.

  • Erdos Erdos
    0

    Regarding your second comment: Sorry, I do not answer questions in other platforms.

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