Finding $\vec{u}.\vec{v}$ Given $|\vec{u}+\vec{v}|=1$ and $|\vec{u}-\vec{v}|=5$

[danago]

If \vec{u},\vec{v} \in R^n, find \vec{u}.\vec{v} given that |\vec{u}+\vec{v}|=1 and that |\vec{u}-\vec{v}|=5.

When i first looked at this i thought i knew how to do it, but i got a bit stuck. I started by finding the dot product:

<br /> \begin{array}{l}<br /> (\overrightarrow u - \overrightarrow v ) \cdot (\overrightarrow u - \overrightarrow v ) = \left| {\overrightarrow u } \right|^2 + \left| {\overrightarrow v } \right|^2 - 2\overrightarrow u \cdot \overrightarrow v \\ <br /> \therefore \left| {\overrightarrow u - \overrightarrow v } \right|^2 = \left| {\overrightarrow u } \right|^2 + \left| {\overrightarrow v } \right|^2 - 2\overrightarrow u \cdot \overrightarrow v \\ <br /> \therefore \overrightarrow u \cdot \overrightarrow v = \frac{{\left| {\overrightarrow u } \right|^2 + \left| {\overrightarrow v } \right|^2 - \left| {\overrightarrow u - \overrightarrow v } \right|^2 }}{2} \\ <br /> \end{array}<br /> <br />

And that's where I am stuck. I am sure its something simple, but i haven't managed to see it

 

[HallsofIvy]

You are given both |\vec{u}-\vec{v}| and |\vec{u}+\vec{v}| but don't seem to have used the value of |\vec{u}+ \vec{v}|. Did you consider looking at (\vec{u}-\vec{v})\cdot(\vec{u}+\vec{v})= |u|^2- |v|^2?

 

[rootX]

I couldn't find relationship between |u|^2-|v|^2

but, continuing your way
(u-v).(u-v) = u^2+v^2-2u.v
(u+v).(u+v) = u^2+v^2+2u.v

now, it's just one step since you know both left side values

 

[danago]

I couldn't find relationship between |u|^2-|v|^2

but, continuing your way
(u-v).(u-v) = u^2+v^2-2u.v
(u+v).(u+v) = u^2+v^2+2u.v

now, it's just one step since you know both left side values

Oh so now i can just solve by eliminating u^2+v^2?

Thanks for the help guys

 

[rootX]

Oh so now i can just solve by eliminating u^2+v^2?

if that answers the question ...