Question

5x-3=15y+7=3-10z equation of line then find direction cosine

Answer 1

Dear Student,

Here is the answer to your query.

$$\begin{gathered} \mathrm{The}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{line}\mathrm{is}5\mathrm{x}-3=15\mathrm{y}+7=3-10\mathrm{z}. \\ \mathrm{The}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{line}\mathrm{in}\mathrm{standard}\mathrm{form}\mathrm{is} \\ \frac{\mathrm{x}-{\displaystyle \frac{3}{5}}}{{\displaystyle \frac{1}{5}}}=\frac{\mathrm{y}-\left(-{\displaystyle \frac{7}{15}}\right)}{{\displaystyle \frac{1}{15}}}=\frac{\mathrm{z}-{\displaystyle \frac{3}{10}}}{-{\displaystyle \frac{1}{10}}} \\ \Rightarrow \frac{\mathrm{x}-{\displaystyle \frac{3}{5}}}{{\displaystyle 3}}=\frac{\mathrm{y}-\left(-{\displaystyle \frac{7}{15}}\right)}{{\displaystyle 1}}=\frac{\mathrm{z}-{\displaystyle \frac{3}{10}}}{-{\displaystyle 2}} \\ \mathrm{The}\mathrm{direction}\mathrm{ratios}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{line}\mathrm{are}3,1,-2. \\ \therefore \mathrm{The}\mathrm{direction}\mathrm{cosines}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{line}\mathrm{are} \\ \frac{3}{\sqrt{9+1+4}},\frac{1}{\sqrt{9+1+4}},\frac{-2}{\sqrt{9+1+4}} \\ \mathrm{or}\frac{3}{\sqrt{14}},\frac{1}{\sqrt{14}},\frac{-2}{\sqrt{14}} \end{gathered}$$