inner product question <,>

Question

Pleace help, that is true or false

Let, $a$, $u$ and $v$ $\in \mathbb{R} $, with $v \neq 0 $, $<,>$ is inner product \begin{eqnarray*} <u,v >&=&<a,v>\\ <u,v>-<a,v>&=&0\\ <u-a,v>&=0& \end{eqnarray*} Then, $u=a$ ???

Thanks

Answer 1

As Daryl has pointed out, it is sufficient that $u-a$ is orthogonal to $v$. In this case the dot product $<u-a,v>$ will be $0$, even if $u-a$ is non zero ($u \ne a)$. An example in 2-dimensional euclidean space is $u=(1, 2)$, $a=(5,8)$, $v=(9,-6)$. In this case, $<u,v>=<a,v>=-3$ but $u \ne a$.