How is the dot product of vectors a and b related to |a+b| and |a-b|?

[Elektrykia]

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. (:

 

[Dick]

|v|^2=v dot v for any vector v. Apply that with v=a+b and v=a-b.

 

[Elektrykia]

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.

 

[Dick]

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)?

 

[Elektrykia]

(a+b) dot (a+b) = a^2 + b^2 + 2ab, yes?

 

[Dick]

(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?

 

[Elektrykia]

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 (: