Question

Find a vector $\overrightarrow{r}$ of magnitude $3\sqrt{2}$ units which makes an angle of $\frac{\mathrm{\pi }}{4}$ and $\frac{\mathrm{\pi }}{2}$ with y and z-axes respectively. [NCERT EXEMPLAR]

Answer 1

Suppose vector $\overrightarrow{r}$ makes an angle α with the x-axis.

Let l, m, n be the direction cosines of $\overrightarrow{r}$. Then,

$l=\cos \alpha ,m=\cos \frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}},n=\cos \frac{\mathrm{\pi }}{2}=0$

Now,

$$\begin{gathered} l^2+m^2+n^2=1 \\ \Rightarrow {\cos }^2\alpha +\frac{1}{2}+0=1 \\ \Rightarrow {\cos }^2\alpha =1-\frac{1}{2}=\frac{1}{2} \\ \Rightarrow \cos \alpha =\pm \frac{1}{\sqrt{2}} \end{gathered}$$

We know that

$$\begin{gathered} \overrightarrow{r}=\left|\overrightarrow{r}\right|\left(l\hat{i}+m\hat{j}+n\hat{k}\right) \\ \therefore \overrightarrow{r}=3\sqrt{2}\left(\pm \frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}j+0\hat{k}\right)\left(\left|\overrightarrow{r}\right|=3\sqrt{2}\right) \\ \Rightarrow \overrightarrow{r}=\pm 3\hat{i}+3\hat{j} \end{gathered}$$