If a line marks angle 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 $2\widehat{\mathrm{ı}}+2\widehat{\mathrm{ȷ}}-3\widehat{\mathrm{k}}$

A line makes ${90}^{\circ },{135}^{\circ },{45}^{\circ }$with x, y and z axes respectively.
Therefore, Direction cosines of the line are $\cos {90}^{\circ },\cos {135}^{\circ }$ and $\cos {45}^{\circ }$
$\Rightarrow$ Direction cosines of the line are $0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$
Therefore, the direction cosines for line marks angle ${90}^{\circ },{135}^{\circ },{45}^{\circ }$with the x, y and z axes respectively are $0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$.

OR

From the given information,  
Vector equation of a line which passes through a point (3,4,5) and parallel to the vector $2\widehat{\mathbf{i}}+2\widehat{\mathbf{j}}-3\widehat{\mathbf{k}}$ is
$\overrightarrow{\mathrm{r}}=3\widehat{\mathbf{i}}+4\widehat{\mathbf{j}}+5\widehat{\mathbf{k}}+\mathrm{\mu }(2\widehat{\mathbf{i}}+2\widehat{\mathbf{j}}-3\widehat{\mathbf{k}})$
Therefore, the vector equation of the line which passes through the point (3, 4, 5) and is parallel to the vector $2\widehat{\mathrm{ı}}+2\widehat{\mathrm{ȷ}}-3\widehat{\mathrm{k}}\text{ is }\overrightarrow{\mathrm{r}}=3\widehat{\mathrm{ı}}+4\widehat{\mathrm{ȷ}}+5\widehat{\mathrm{k}}+\mathrm{\mu }(2\widehat{\mathrm{ı}}+2\widehat{\mathrm{ȷ}}-3\widehat{\mathrm{k}})$