Answer
$ \sqrt 2$
Work Step by Step
$u=-\frac{1}{\sqrt 2}i+\frac{1}{\sqrt 2}j$
$v=-\frac{1}{\sqrt 2}i-\frac{1}{\sqrt 2}j$
$u+v=-\frac{2}{\sqrt 2}i$
let $w= |w|i$
$u+v+w=0$ (given)
$-\frac{2}{\sqrt 2}i+ |w|i=0$
$|w|i=\frac{2}{\sqrt 2}i$
Hence, $|w|=\frac{2}{\sqrt 2} \times \frac{\sqrt 2}{\sqrt 2}=\sqrt 2$
