If $(2 \vec{a}+3 \vec{b}) \cdot \vec{c}=1670$, then $|\vec{c}|^2$ is equal to :
$ (\vec{a}+\vec{b}) \times \vec{c}+(-2 \vec{a}+3 \vec{b}) \times \vec{c}=0 $
$ \Rightarrow(\vec{a}+\vec{b})-2 \vec{a}+3 \vec{b}) \times \vec{c}=0 $
$ \Rightarrow \vec{c}=\lambda(4 \vec{b}-\vec{a}) $
$ \Rightarrow=\lambda(44 \hat{i}-4 \hat{j}+4 \hat{k}-4 \hat{i}+\hat{j}-\hat{k}) $
$ =\lambda(40 \hat{i}-3 \hat{j}+3 \hat{k})$
Now
$ (8 \hat{i}-2 \hat{j}+2 \hat{k}+33 \hat{i}-3 \hat{j}+3 \hat{k}) \cdot \lambda(40 \hat{i}-3 \hat{j}+3 \hat{k})=1670 $
$ \Rightarrow(41 \hat{i}-5 \hat{j}+5 \hat{k}) \cdot(40 \hat{i}-3 \hat{j}+3 \hat{k}) \times \lambda=1670) $
$ \Rightarrow(1640+15+15) \lambda=1670 \Rightarrow \lambda=1 $
$ \text { so } \overrightarrow{\mathrm{c}}=40 \hat{i}-3 \hat{j}-3 \hat{k} $
$ \Rightarrow|\overrightarrow{\mathrm{c}}|^2=1600+9+9=1618$
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