Given that $\overrightarrow{\mathrm{u}}=\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+3\widehat{\mathrm{k}};\overrightarrow{\mathrm{v}}=2\widehat{\mathrm{i}}+\widehat{\mathrm{j}}+4\widehat{\mathrm{k}};\overrightarrow{\mathrm{w}}=\widehat{\mathrm{i}}+3\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\text{ and }(\overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{R}}-15)\widehat{\mathrm{i}}+(\overrightarrow{\mathrm{v}}\cdot \overrightarrow{\mathrm{R}}-30)\widehat{\mathrm{j}}+(\overrightarrow{\mathrm{w}}\cdot \overrightarrow{\mathrm{R}}-20)\widehat{\mathrm{k}}=\overrightarrow{0}$ Then find the greatest integer less than or equal to $\left|\overrightarrow{\mathrm{R}}\right|$

$\text{ Let }\overrightarrow{R}=x\widehat{i}+\widehat{y}\widehat{j}+z\widehat{k}$

$\begin{array}{r}\overrightarrow{u}=\widehat{i}-2\widehat{j}+3\widehat{k};\overrightarrow{v}=2\widehat{i}+\widehat{j}+4\widehat{k};\overrightarrow{w}=\widehat{i}+3\widehat{j}+3\widehat{k}\\ (\overrightarrow{u}\cdot \overrightarrow{R}-15)\widehat{i}+(\overrightarrow{v}\cdot \overrightarrow{R}-30)\widehat{j}+(\overrightarrow{w}\cdot \overrightarrow{R}-25)\widehat{k}=\overrightarrow{0}\end{array}$

so,

$\begin{array}{r}\overrightarrow{\mathrm{u}}\cdot \overrightarrow{\mathrm{R}}=15\Rightarrow \mathrm{x}-2\mathrm{y}+3\mathrm{z}=15---\left(\mathrm{i}\right)\\ \overrightarrow{\mathrm{v}}\cdot \overrightarrow{\mathrm{R}}=30\Rightarrow 2\mathrm{x}+\mathrm{y}+4\mathrm{z}=30---\left(\mathrm{ii}\right)\\ \overrightarrow{\mathrm{w}}\cdot \overrightarrow{\mathrm{R}}=25\Rightarrow \mathrm{x}+3\mathrm{y}+3\mathrm{z}=25---\left(\mathrm{iii}\right)\end{array}$

Solving, we get 

$\begin{array}{r}\mathrm{x}=4\\ \mathrm{y}=2\\ \mathrm{z}=5\end{array}$