Question

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

Answer 1

Suppose vector \[\vec{r}\] makes an angle α with the x-axis. 
Let l, m, n be the direction cosines of \[\vec{r}\].
Then, \[l = \cos\alpha, m = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}, n = \cos\frac{\pi}{2} = 0\] 
Now,

\[l^2 + m^2 + n^2 = 1\]

\[ \Rightarrow \cos^2 \alpha + \frac{1}{2} + 0 = 1\]

\[ \Rightarrow \cos^2 \alpha = 1 - \frac{1}{2} = \frac{1}{2}\]

\[ \Rightarrow \cos\alpha = \pm \frac{1}{\sqrt{2}}\]
We know that  \[\vec{r} = \left| \vec{r} \right|\left( l \hat{i} + m \hat{j} + n \hat{k} \right)\]
\[ \therefore \vec{r} = 3\sqrt{2}\left( \pm \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}}j + 0 \hat{k} \right) \left( \left| \vec{r} \right| = 3\sqrt{2} \right)\]
\[ \Rightarrow \vec{r} = \pm 3 \hat{i} + 3 \hat{j}\]