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$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{direction} \mathrm{cosine} \mathrm{of} \mathrm{any} \mathrm{line} \mathrm{are}: \\ \mathrm{cos\alpha },\mathrm{cos\beta },\mathrm{cos\gamma } \mathrm{which} \\ \mathrm{cos\alpha }=\frac{2}{\sqrt{3}} \\ \mathrm{But} \frac{2}{\sqrt{3}}>1 \mathrm{and} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{cosine} \mathrm{is} 1. \\ \mathrm{Hence} \mathrm{cos\alpha }=\frac{2}{\sqrt{3}} \mathrm{is} \mathrm{not} \mathrm{possible}. \\ \Rightarrow \frac{2}{\sqrt{3}},-\frac{2}{\sqrt{3}},-\frac{1}{\sqrt{3}} \mathrm{cannot} \mathrm{be} \mathrm{direction} \mathrm{cosines}. \end{gathered}$$

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