Question

Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3).

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{direction}\mathrm{cosines}\mathrm{of}\mathrm{the}\mathrm{line}\mathrm{passing}\mathrm{through}\mathrm{two}\mathrm{points}\mathrm{P}\left({\mathrm{x}}_{\mathit{1}}\mathit{,}\mathit{}{\mathrm{y}}_{\mathit{1}}\mathit{,}\mathit{}{\mathrm{z}}_{\mathit{1}}\right)\mathrm{and}\mathrm{Q}\left({\mathrm{x}}_2,{\mathrm{y}}_2,{\mathrm{z}}_2\right)\mathrm{are}\frac{{\mathrm{x}}_2-{\mathrm{x}}_1}{\mathrm{PQ}},\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{\mathrm{PQ}},\frac{{\mathrm{z}}_2-{\mathrm{z}}_1}{\mathrm{PQ}}. \\ \mathrm{Here}, \\ \mathrm{PQ}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2+{\left({\mathrm{z}}_2-{\mathrm{z}}_1\right)}^2} \\ \mathrm{P}=\left(-2,4,-5\right) \\ \mathrm{Q}=\left(1,2,3\right) \\ \therefore \mathrm{PQ}=\sqrt{{\left[1-\left(-2\right)\right]}^2+{\left(2-4\right)}^2+{\left[3-\left(-5\right)\right]}^2}=\sqrt{77} \\ \mathrm{Thus},\mathrm{the}\mathrm{direction}\mathrm{cosines}\mathrm{of}\mathrm{the}\mathrm{line}\mathrm{joining}\mathrm{two}\mathrm{points}\mathrm{are}\frac{1-\left(-2\right)}{\sqrt{77}},\frac{2-4}{\sqrt{77}},\frac{3-\left(-5\right)}{\sqrt{77}},\mathrm{i}.\mathrm{e}.\frac{3}{\sqrt{77}},\frac{-2}{\sqrt{77}},\frac{8}{\sqrt{77}}. \end{gathered}$$