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I am looking for an intuitive explanation of why the dot product written as

$x \cdot y=\sum x_i y_i = x_1y_1 + x_2y_2 + x_3y_3 + x_4y_4 + ......$

has anything to do with the cosine of the angle between $x$ and $y$. I dont want a formal proof but an explanation of how the angle or projection fits into this summation. I can't bridge the summation and these concepts.

I am looking for a conceptual bridge between the length of the projection and the above equation. It is baffling me that we can incorporate cosine in such a linear form.

Here is a comment that I did not understand and would like an expansion on:

the relation to the cosine (law) may be not the best way to see that. I like to think of it in terms of orthonormal basis. Given an orthonormal basis, any vector is simply $\sum_{i}\left(u \cdot v_{i}\right) \cdot v_{i}$ by a trivial computation which has nothing to do with CW or the law of cosines. This trivially shows that the dot product gives the components of a vector. When teaching these things I like to use the law of cosines to motivate the definition of the standard inner product. Then CS is shown to relate to a law of cosines (kind of) for any inner product.

Source: Intuition for the Cauchy-Schwarz inequality in the comments of the ticked answer

Rahul Deora
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  • Ultimately it's the cosine rule from trigonometry. – Angina Seng Jul 08 '19 at 08:21
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    In physics and even in mathematics, the dot product is kind of projection of one vector onto the another. So (for 2-D at least) one can easily see why a cosine should come. It is just expressing one vector(the base) in terms of the other vector (the hypotenuse) or vice-versa. – Rick Jul 08 '19 at 08:28
  • I am asking how can the projection be seen from the summation term @Rick – Rahul Deora Jul 08 '19 at 08:49
  • "has anything to do with the cosine of the angle between x and y" I commented seeing this. Nevertheless you will get an answer soon. – Rick Jul 08 '19 at 09:08
  • 3Blue1Brown made an excellent video on the dot product in his Essence of Linear Algebra series. It may help you understand the connection – csch2 Jul 08 '19 at 16:30

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