The Dot Product Expansion

  • Thread starter Elektrykia
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  • #1
Elektrykia
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Homework Statement


Show that the dot product of vectors a and b is equal to 1/4|a+b|^2 - 1/4|a-b|^2


Homework Equations


a dot b = |a||b|cos(theta)
a dot b = a1b1 + a2b2 + ...


The Attempt at a Solution


I've tried using the combination of the cosine law and those two above dot product equations and I have gotten no where. Any help would be greatly appreciated. (:
 

Answers and Replies

  • #2
Dick
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|v|^2=v dot v for any vector v. Apply that with v=a+b and v=a-b.
 
  • #3
Elektrykia
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I've tried that, maybe there's something I'm missing but I do v=a+b and u=a-b and get them to a point where i can expand them out and I get 1/4ab as the answer.
 
  • #4
Dick
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The ab=a dot b is right. The 1/4 isn't right. What do you get for the expansion of (a+b) dot (a+b)?
 
  • #5
Elektrykia
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(a+b) dot (a+b) = a^2 + b^2 + 2ab, yes?
 
  • #6
Dick
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(a+b) dot (a+b) = a^2 + b^2 + 2ab, yes?

Yes. 'ab' means 'a dot b', correct? And (a-b) dot (a-b)? And the difference between the two?
 
  • #7
Elektrykia
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Haha, oh wow, I looked over that like four times.
For some reason ab looked so wrong to me when I got it, I understand it now.

Thanks very much (:
 

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