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Why is it that the dot product of two vectors $u_1=(a, b, c)$ and $u_2=(x, y, z)$ plays out like this: $u_1 \cdot u_2 = ax+by+cz$. I have seen and understand why this is the case for 2D but why is it the case for 3D? I though the dot product is $|u||v|\cos \theta$? This would result in $\sqrt{a^2+b^2+c^2} \sqrt{x^2+y^2+z^2} \cos \theta$. Are they the same?

Edit: Again, I have seen the proof for the second dimension but I have not seen any proof for the equation $u_1 \cdot u_2 = ax+by+cz$. Can anybody guide me? Also, does this apply in all dimensions?

linker
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    "I have seen and understand why this is the case for 2D but why is it the case for 3D?" The proofs are nearly identical. If you understand one, then you really ought to understand both. – Arthur Mar 03 '21 at 15:53
  • @HansLundmark No it does not because it does not include the third dimension. I did not ask for 2D. In fact, I specifically did not ask for 2D definition. – linker Mar 03 '21 at 16:07
  • @Arthur I have a lot going on right now so if I cannot see something as quickly as you can then I am sorry. Your comment about them being nearly identical does not explain their difference nor answer my question. – linker Mar 03 '21 at 16:09
  • Oh, sorry... That question asked for the 3D case, but I didn't notice that the answers only did the 2D case. Anyway, you should find the answer somewhere among all the umpteen other duplicates if you follow the links. – Hans Lundmark Mar 03 '21 at 16:10
  • Here, for example: https://math.stackexchange.com/questions/2380217/why-are-the-two-dot-product-definitions-equal – Hans Lundmark Mar 03 '21 at 16:11
  • I didn't intend to answer your question. That's why I wrote a comment and not an answer. However, I did want to try to point you in the direction of reading up on why it works in 2D, as you claim to understand, and then just... add another coordinate. See what happens. Try to put words to where exactly you can't really understand what's going on. That would make it easier for us to help you in particular. – Arthur Mar 03 '21 at 17:25
  • If you understand cosine rule in plane Euclidean geometry and the definition of angle in all dimensions, then you should understand why this works. – user10354138 Mar 03 '21 at 18:20
  • @user10354138 key word: should. I do not. – linker Mar 03 '21 at 20:33
  • @linker Just apply the cosine rule to the triangle $OU_1U_2$. – user10354138 Mar 03 '21 at 21:00
  • This is in every sense the definition of the dot product. – K.defaoite Mar 06 '21 at 04:09

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