Question

A vector $\overrightarrow{r}$ is inclined to x-axis at 45° and y-axis at 60°. If $\left|\overrightarrow{r}\right|$ = 8 units, find $\overrightarrow{r}$.

Answer 1

Here, $\overrightarrow{r}$ makes an angle $45°$ with $OX$ and $60°$ with $OY$. So,
$l=\cos 45°=\frac{1}{\sqrt{2}}\mathrm{and}m=\cos 60°=\frac{1}{2}$
$$\begin{gathered} \mathrm{Now},l^2+m^2+n^2=1 \\ \Rightarrow \frac{1}{2}+\frac{1}{4}+n^2=1 \\ \Rightarrow n^2=\frac{1}{4} \\ \Rightarrow n=\pm \frac{1}{2} \end{gathered}$$
Therefore,
$$\begin{gathered} \overrightarrow{r}=\left|\overrightarrow{r}\right|\left(l\hat{i}+m\hat{j}+n\hat{k}\right) \\ =8\left(\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{2}\hat{j}\pm \frac{1}{2}\hat{k}\right) \\ =4\left(\sqrt{2}\hat{i}+\hat{j}\pm \hat{k}\right) \end{gathered}$$