What is $\langle 3u+2v , -u+4v\rangle$ Given $\lVert u\rVert$ , $\lVert v\rVert$ , and $\langle u,v\rangle$? What is $\langle 3u+2v , -u+4v\rangle$ Given $\lVert u\rVert$ , $\lVert v\rVert$ , and $\langle u,v\rangle$?
\begin{align}
    \lVert u\rVert &amp;= 3\\
    \lVert v\rVert &amp;= 5\\
    \langle u,v\rangle &amp;= -4
\end{align}
Well, to solve this I thought I'd break down certain elements.

I know that...
\begin{gather}
 \langle u,v\rangle = \mathbf{u}\cdot\mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 + \cdots + u_nv_n = -4\\
    \langle u+v , u+v\rangle = \langle u,u\rangle + \langle u,v\rangle + \langle v,u\rangle + \langle v,v\rangle
\end{gather}
But I don't know what $\langle 3u+2v , -u+4v\rangle$ actually is...

Is it 
$$
    \langle 3u,-u\rangle + \langle 3u,4v\rangle + \langle 2v,-u\rangle + \langle 2v,4v\rangle \mbox{?}
$$
Hints.$\def\v#1{{\bf#1}}$


Expand $(3\v u+2\v v , -\v u+4\v v)$ pretty much as you did for $(\v u+\v v,\v u+\v v)$.
You are given $(\v u,\v v)$: how does this help you to find $(\v v,\v u)$?
You are given $\|\v u\|$ and $\|\v v\|$: what does this tell you about the other terms in your expansion?
$\langle 3u+2v,-u+4v\rangle    =\langle 3u,-u\rangle +\langle 3u,4v\rangle +\langle 2v,-u\rangle +\langle 2v,4v\rangle$
\begin{align}
          &amp;=-3\langle u,u\rangle +12\langle u,v\rangle -2\langle v,u\rangle +8\langle v,v\rangle\\
              &amp;=-3\cdot 9 + 12\cdot -4 - 2\cdot 4 + 8\cdot 25
\end{align}
since $\lVert u\rVert = 3$, $\lVert v\rVert = 5$, $\langle u,v\rangle = -4$ and  $\lVert u\rVert^2 =\langle u,u\rangle$