I need some help figuring out how to work through this problem.

Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$.

Sorry, forgot to include my work so far:

I decided to ignore the 1/4 and deal with it later once I had a better understanding of the question.

$= ||u+v||^2 - ||u-v||^2$

$= (u+v)(u+v) - (u-v)(u-v)$

$= u(u+v) + v(u+v) - u(u-v) + v(u-v)$

$= uu + uv + uv + vv - uu + uv + uv - vv$

$u \cdot v= 3(uv)$

This is as far as I've gotten, not sure if I'm on the right track or where to go next.

  • 3
    $\begingroup$ Did you try anything at all? If you expand the right hand side, this is just a couple lines of algebra. $\endgroup$
    – user296602
    Feb 8 '16 at 5:26
  • $\begingroup$ What do you already know about the dot product, and what have you tried? $\endgroup$ Feb 8 '16 at 5:26
  • $\begingroup$ @T.Bongers Sorry mate, I intended on including what I've gotten so far but my daughter woke up mid-post. Figured I'd ask and get some input while I put her back to sleep :). $\endgroup$ Feb 8 '16 at 6:16
  • 3
    $\begingroup$ @TalenKylon You have $4 u \cdot v$, count up your terms again. $\endgroup$
    – mattos
    Feb 8 '16 at 6:53
  • 2
    $\begingroup$ @TalenKylon You don't have $u \cdot v = 4 u \cdot v$. You have $$\lvert \lvert u + v \rvert \rvert^{2} + \lvert \lvert u - v \rvert \rvert^{2} = 4 u \cdot v$$ Now just divide both sides by $4$ and you have the result you required. $\endgroup$
    – mattos
    Feb 9 '16 at 0:35

Here is a start

$$||u+v||^2= \langle u+v, u+v \rangle= \langle u, u \rangle+\dots \,. $$

Do the same with the other and multiply both eqs and by $\frac{1}{4}$ and subtract. See my answer.


$$\langle u, v\rangle = u.v \,. $$

  • 2
    $\begingroup$ You might want to use \langle and \rangle for the inner product notation. $\endgroup$
    – mattos
    Feb 8 '16 at 5:43
  • 1
    $\begingroup$ @Mattos: you are right! Thank you! $\endgroup$ Feb 8 '16 at 5:45

Remember $\vec{x} \cdot \vec{x} = \left \| x \right \|^2 $. That should be really helpful, in my mind.

And as @T.Bongers said, when working with identities like this, begin with the harder side. Try to "simplify" it (or make it a big more complex before having things cancel!) into the more basic side.


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