Find all the possible vectors of magnitude  $5\sqrt{3}$ which are equally inclined to the coordinate axes.

From the given problem vectors are equally inclined to the coordinate axes,
Let’s assume direction cosines are (a, b, c)
Hence, a=b=c (∵ vectors are equally inclined to the coordinate axes)
So, the direction cosines are (a, a, a)
Let’s find the magnitude,
$\begin{array}{l}\therefore \sqrt{{\mathrm{a}}^2+{\mathrm{a}}^2+{\mathrm{a}}^2}=1\\ \therefore \sqrt{3{\mathrm{a}}^2}=1\\ \therefore \mathrm{a}=\pm \frac{1}{\sqrt{3}}\end{array}$
Hence, directional vector is
$\pm \frac{1}{\sqrt{3}}(\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{\mathrm{k}})$
For required vector magnitude is $5\sqrt{3}$
$\begin{array}{l}\pm 5\sqrt{3}\cdot \frac{1}{\sqrt{3}}(\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{\mathrm{k}})\\ \Rightarrow \pm 5(\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{\mathrm{k}})\end{array}$
Therefore, required vector is $\pm 5(\widehat{\mathrm{ı}}+\widehat{\mathrm{ȷ}}+\widehat{\mathrm{k}})$