# How to show $\vec{a} \cdot \vec{b}=\frac{1}{4}\left(\Vert{\vec{a}+\vec{b}\Vert^2}-\Vert{\vec{a}-\vec{b}\Vert^2}\right)$? [duplicate]

The Homework Exercise I am working on, is:

Let $\overrightarrow{a}, \overrightarrow{b}$ be vectors. Show that $\overrightarrow{a} \cdot \overrightarrow{b}=\frac{1}{4}\left(\Vert{\overrightarrow{a}+\overrightarrow{b}\Vert^2}-\Vert{\overrightarrow{a}-\overrightarrow{b}\Vert^2}\right)$.

Things I tried:

1. Using the property $\Vert \overrightarrow{u}\Vert^2=\overrightarrow{u} \cdot \overrightarrow{u}$, I tried to make $\Vert{\overrightarrow{a}+\overrightarrow{b}\Vert^2} =\left(\overrightarrow{a}+\overrightarrow{b} \right) \cdot \left(\overrightarrow{a}+\overrightarrow{b} \right)$ and $\Vert{\overrightarrow{a}-\overrightarrow{b}\Vert^2} =\left(\overrightarrow{a}-\overrightarrow{b} \right) \cdot \left(\overrightarrow{a}-\overrightarrow{b} \right)$

I'm not sure if that was a correct step, but then I substituted into the main equation to get $$\overrightarrow{a} \cdot \overrightarrow{b} = \frac{1}{4}\left( (\overrightarrow{a}+\overrightarrow{b} ) \cdot (\overrightarrow{a}+\overrightarrow{b} ) - (\overrightarrow{a}-\overrightarrow{b} ) \cdot (\overrightarrow{a}-\overrightarrow{b} ) \right)$$

Am not sure where to go from here, or if I'm even heading in the right direction...

## marked as duplicate by Jack, MrYouMath, Namaste linear-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 17 '17 at 16:03

Use the fact that $$\vec a\cdot (\vec b + \vec c) = \vec a \cdot \vec b + \vec a\cdot \vec c$$