Question

Prove the polarization identity,

${\left|\left|u+v\right|\right|}^2-{\left|\left|u-v\right|\right|}^2=4\left(u·v\right)$

Short answer

${\left|\left|u+v\right|\right|}^2-{\left|\left|u-v\right|\right|}^2=4\left(u·v\right)$

Step-by-step solution

Step 1. Given information

We have to prove the polarization identity

${\left|\left|u+v\right|\right|}^2-{\left|\left|u-v\right|\right|}^2=4\left(u·v\right)$

Step 2. Proof

We can write ${\left|\left|u+v\right|\right|}^2$as $\left(u+v\right)·\left(u+v\right)$

and ${\left|\left|u-v\right|\right|}^2$as $\left(u-v\right)·\left(u-v\right)$

Now,

role="math" localid="1646800338985" $$\begin{gathered} {\left|\left|u+v\right|\right|}^2-{\left|\left|u-v\right|\right|}^2 \\ =\left(u+v\right)·\left(u+v\right)-\left(u-v\right)·\left(u-v\right) \\ =\left(u·u+u·v+v·u+v·v\right)-\left(u·u-u·v-v·u+v·v\right) \\ =u·u+u·v+v·u+v·v-u·u+u·v+v·u-v·v \\ =2\left(u·v\right)+2\left(v·u\right) \end{gathered}$$

Using commutative property, $\left(u·v\right)=\left(v·u\right)$

So,

$$\begin{gathered} 2\left(u·v\right)+2\left(v·u\right) \\ =4\left(u·v\right) \end{gathered}$$

Therefore,${\left|\left|u+v\right|\right|}^2-{\left|\left|u-v\right|\right|}^2=4\left(u·v\right)$