If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
                                                             OR
Find the vector equation of the line which passes through the point (3, 4, 5) and is parallel to the vector $2\mathrm{î} + 2\mathrm{ĵ} – 3\mathrm{k̂} .$

$$\begin{gathered} \mathrm{Let} \mathrm{direction} \mathrm{cosines} \mathrm{of} \mathrm{the} \mathrm{line} \mathrm{be} \mathrm{l}, \mathrm{m} \mathrm{and} \mathrm{n}, \\ \mathrm{Given}, \mathrm{\alpha }=90°, \mathrm{\beta }=135° \mathrm{and} \mathrm{\gamma }=45° \\ \mathrm{Then}, \mathrm{l}=\cos \mathrm{\alpha }=\cos 90°=0 \\ \mathrm{m}=\cos \mathrm{\beta }=\cos 135°=\frac{-1}{\sqrt{2}} \\ \mathrm{and} \mathrm{n}=\cos \mathrm{\gamma }=\cos 45°=\frac{1}{\sqrt{2}} \\ \mathrm{Hence}, \mathrm{the} \mathrm{direction} \mathrm{cosines} \mathrm{of} \mathrm{a} \mathrm{line} \mathrm{are} 0,\frac{-1}{\sqrt{2}}\mathrm{and}\frac{1}{\sqrt{2}} \end{gathered}$$

OR

$$\begin{gathered} \mathrm{Equation} \mathrm{of} \mathrm{a} \mathrm{line} \mathrm{passing} \mathrm{through} \mathrm{a} \mathrm{point} \mathrm{with} \mathrm{position} \mathrm{vector} \overrightarrow{\mathrm{a}} \mathrm{and} \mathrm{parallel} \mathrm{to} \mathrm{vector} \overrightarrow{\mathrm{b}} \mathrm{is} \\ \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\mathrm{\lambda }\overrightarrow{\mathrm{b}} \\ \mathrm{Since}, \mathrm{line} \mathrm{passes} \mathrm{through} (3,4,5) \\ \therefore \overrightarrow{\mathrm{b}}=3\hat{\mathrm{i}}+4\hat{\mathrm{j}}+5\hat{\mathrm{k}} \\ \mathrm{Since}, \mathrm{line} \mathrm{is} \mathrm{parallel} \mathrm{to} 2\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}} \\ \therefore \overrightarrow{\mathrm{b}}=2\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}} \\ \mathrm{Equation} \mathrm{of} \mathrm{line} \mathrm{is} \overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}}+\mathrm{\lambda }\overrightarrow{\mathrm{b}}, \mathrm{i}.\mathrm{e}., \\ \overrightarrow{\mathrm{r}}=(3\hat{\mathrm{i}}+4\hat{\mathrm{j}}+5\hat{\mathrm{k}})+\mathrm{\lambda }\left(2\hat{\mathrm{i}}+2\hat{\mathrm{j}}-3\hat{\mathrm{k}}\right) \\ \mathrm{which} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{vector} \mathrm{equation}. \end{gathered}$$