Answer
$|\vec{w}| = \sqrt 2$
Work Step by Step
The magnitudes of $\vec{u}$ and $\vec{v}$ are $|\vec{u}|=1$ and $|\vec{v}|=1$.
We have that $\vec{u} + \vec{v} + \vec{w} = \vec{0}$. What this means is that if we travel on $\vec{u}$, then $\vec{v}$, then $\vec{w}$, we arrive at the same location. This makes for a vector diagram that forms a triangle, as shown here.
For a right-angle triangle with two sides lengths of 1, the final side length, using the Pythagorean theorem, is $\sqrt{2}$.
$c=\sqrt{1^2+1^2}$
This final side length corresponds to the magnitude of $\vec{w}$.
