question_answer

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

Answer:

Let \[({{x}_{1}},\,\,{{y}_{1}},\,\,{{z}_{1}})\equiv (-\,2,\,\,4,\,\,-\,5)\] and \[({{x}_{2}},\,\,{{y}_{2}},\,\,{{z}_{2}})\equiv (1,\,\,2,\,\,3)\] DR's of the line are \[1-(-\,2),\,\,2-4,\,\,3-(-\,5)=3,\,\,-\,2,\,\,8\] \[[\because \,\,DR'\,\,\text{of}\,\,\text{the}\,\,\text{are}\,\,{{x}_{2}}-\,x,\,{{y}_{2}}-{{y}_{1}}\,\,and\,\,{{z}_{2}}-{{z}_{1}}]\] \[\therefore \] DC's are \[\frac{3}{\sqrt{{{(3)}^{2}}+{{(-\,2)}^{2}}+{{(8)}^{2}}}},\] \[\frac{-\,2}{\sqrt{{{(3)}^{2}}+{{(-\,2)}^{2}}+{{(8)}^{2}}}},\] \[\frac{8}{\sqrt{{{(3)}^{2}}+{{(-\,2)}^{2}}+{{(8)}^{2}}}}\] \[=\frac{3}{\sqrt{9+4+64}},\] \[\frac{-\,2}{\sqrt{9+4+64}},\] \[\frac{8}{\sqrt{9+4+64}}\] \[=\frac{3}{\sqrt{77}},\] \[\frac{-\,2}{\sqrt{77}},\] \[\frac{8}{\sqrt{77}}\]