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If we can find a vector $\vect{w_1}$ that is a linear combination of $\vect{u}$ and $\vect{v}$, then $\spn{\set{\vect{u}, \vect{v}, \vect{w}_1}}$ will be the same set as $\spn{\set{\vect{u}, \vect{v}}}$. Thus, $\vect{w}_1$ can be any linear combination of $\vect{u}$ and $\vect{v}$. One such example is
$\vect{w}_1 = 3\vect{u} - \vect{v} = \colvector{1\\11\\-7}$.
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Now we are looking for a vector $\vect{w}_2$ that cannot be written as a linear combination of $\vect{u}$ and $\vect{v}$. How can we find such a vector? Any vector that matches two components but not the third of any element of $\spn{\set{\vect{u}, \vect{v}}}$ will not be in the span. (Why?) One such example is
$\vect{w}_2 = \colvector{4\\-4\\1}$
(which is nearly $2\vect{v}$, but not quite).