Question

A vector \[\vec{r}\] is inclined to -axis at 45° and y-axis at 60°.  If \[|\vec{r}|\] = 8 units, find \[\vec{r}\].

Answer 1

Here, \[\vec{r}\] makes an angle \[45^{\circ}\] with OX and \[60^{\circ}\] with OY.
So,
\[l = \cos 45^{\circ} = \frac{1}{\sqrt{2}}\text{ and }m = \cos 60^{\circ} = \frac{1}{2}\]

\[\text{ Now, }l^2 + m^2 + n^2 = 1\]

\[ \Rightarrow \frac{1}{2} + \frac{1}{4} + n^2 = 1\]

\[ \Rightarrow n^2 = \frac{1}{4}\]

\[ \Rightarrow n = \pm \frac{1}{2}\]
Therefore, 
\[\vec{r} = \left| \vec{r} \right| \left( l \hat{i} + m \hat{j} + n \hat{k} \right)\]
\[ = 8 \left( \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{2} \hat{j} \pm \frac{1}{2} \hat{k} \right)\]
\[ = 4 \left( \sqrt{2} \hat{i} + \hat{j} \pm \hat{k} \right)\]