$\vec{a} \times \vec{c}+\vec{b}=0$
$ \Rightarrow \left| {\begin{array}{*{20}{c}}
{\hat i}&{\hat j}&{\hat k}\\
1&{ - 1}&0\\
x&y&z
\end{array}} \right|$
$+(\hat{i}+\hat{\bar{j}}+\hat{k})=0$
$\hat{i}(-z)-\hat{j}(z)+\hat{k}(y+x)$
$\Rightarrow 1-z=0 \Rightarrow z=1$
$\text { Also } x+y=-1, \text { and } \vec{a} \cdot \vec{c}=4 \Rightarrow x-y=4$
$\Rightarrow x=\frac{3}{2}, y=\frac{5}{2}$
$\therefore|\overrightarrow{\mathrm{c}}|^{2}=x^{2}+y^{2}+z^{2}$
$=\frac{9}{4}+\frac{25}{4}+1=\frac{38}{4}=\frac{19}{2}$
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