Suppose V is a two-dimensional vector space.
According to Harris' "Algebraic Geometry", p.118 the vectors in $\mathbb{P}(\operatorname{Sym}^2(V))$ of the form $v\cdot v$ are supposed to be exactly the elements of $v_2(\mathbb{P}^1)$, which is the image of the quadratic Veronese map.
However, I fail to see this correspondence. If I take basis vectors $v_1, v_2$ of V and suppose that $v=\lambda_1v_1 + \lambda_2v_2$, wouldn't $v\cdot v$ just be equal to
$\lambda_1^2(v_1\cdot v_1) + 2\lambda_1\lambda_2(v_1\cdot v_2)+\lambda_2^2(v_2\cdot v_2)$ in $\operatorname{Sym}^2(V)$,
and thus correspond to $[\lambda_1^2:2\lambda_1\lambda_2:\lambda_2^2]$ in homogeneous coordinates (which is not in the image of $v_2$)?
I assume that I have either just miscalculated or severely misunderstood some central concept. Either way, I'd be grateful for some help.