Claim: $(\forall u\in \mathbb{R}^2)$ $(\nexists(\delta,v)\in(\mathbb{R}, \mathbb{R}^2))$ such that
$uu'+vv'=\delta \begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$.
That is, for any vector $u$ of dimension 2 there does not exist a combination of a scalar $\delta$ and another vector $v$ of dimension 2 s.t. the above equation holds.
Attempted Proof:
Proof by contradiction: For given $u$ assume there exists $\delta$ and $v$ such that the equation holds. Then
$\begin{pmatrix} u_1u_1 & u_1u_2 \\ u_2u_1 & u_2u_2 \end{pmatrix}$ + $\begin{pmatrix} v_1v_1 & v_1v_2 \\ v_2v_1 & v_2v_2 \end{pmatrix}=\begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$ $\implies u_2^2+v_2^2=0 \implies v_2^2=-u_2^2<0$ which contradicts $v\in\mathbb{R}^2$.