Question

A vector $\overrightarrow{r}$ is inclined at equal acute angles to x-axis, y-axis and z-axis. If $\left|\overrightarrow{r}\right|$ = 6 units, find $\overrightarrow{r}$.

Answer 1

Suppose, vector $\overrightarrow{r}$ makes an angle $\alpha$ with each of the axis $OX,OY$ and $OZ$. Then, its direction cosines are $l=\cos \alpha ,m=\cos \alpha$ and $n=\cos \alpha$ i.e. $l=m=n.$

$$\begin{gathered} \mathrm{Now},l^2+m^2+n^2=1 \\ \Rightarrow l^2+l^2+l^2=1 \\ \Rightarrow 3l^2=1 \\ \Rightarrow l^2=\frac{1}{3} \\ \Rightarrow l=\pm \frac{1}{\sqrt{3}} \\ \mathrm{Since},\overrightarrow{r}\mathrm{makes}\mathrm{acute}\mathrm{angle}\mathrm{with}\mathrm{the}\mathrm{axis}. \\ \mathrm{Hence},\mathrm{we}\mathrm{take}\mathrm{only}\mathrm{positive}\mathrm{value}. \end{gathered}$$
Therefore,

$$\begin{gathered} \overrightarrow{r}=\left|\overrightarrow{r}\right|\left(l\hat{i}+m\hat{j}+n\hat{k}\right) \\ \overrightarrow{r}=6\left(\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}\right) \\ =2\sqrt{3}\left(\hat{i}+\hat{j}+\hat{k}\right) \end{gathered}$$