A Householder transformation is a _reflection_ about a plane. That is, under a suitable orthonormal basis $\\{w_1,w_2,\ldots,w_n\\}$, a Householder transformation $H$ is simply the linear transformation defined by $Hw_1=-w_1$ and $Hw_j=w_j$ for $j>1$. It follows that every Householder transformation is a unitary matrix that preserves norms of vectors. So, your statement is _not_ true if $u$ and $v$ have different lengths.
Now, given two vectors $u$ and $v$ of the same norm, let $P$ be a two-dimensional plane containg them. Let $w_2\in P$ be a unit vector that is an angle bisector of $u$ and $v$ (i.e. $w_2=\frac{u+v}{\|u+v\|}$ if $u,v$ are nonzero, or $w_2$ is any unit vector if $u=v=0$) and $w_1\in P$ be a unit vector that is orthogonal to $w_2$. Extend $\\{w_1,w_2\\}$ to an orthonormal basis $\\{w_1,w_2,\ldots,w_n\\}$. Then the Householder transformation we previously defined will map $u$ to $v$. You may draw a picture to visualize this.