Question

The Cartesian equations of the straight line passing through the point (2, 1, –1) and making angles $\frac{\mathrm{\pi }}{3},\frac{\mathrm{\pi }}{3}\mathrm{and}\frac{\mathrm{\pi }}{4}$with the positive directions of the coordinate axes are _____________.

Answer 1

Since straight line makes angle $\frac{\mathrm{\pi }}{3},\frac{\mathrm{\pi }}{3}\mathrm{and}\frac{\mathrm{\pi }}{4}$ with positive direction of co-ordinate axes
∴ direction cosines are given by

$$\begin{gathered} l=\cos \frac{\mathrm{\pi }}{3},m=\cos \frac{\pi }{3},n=\cos \frac{\mathrm{\pi }}{4} \\ \mathrm{i}.\mathrm{e}.l=\frac{1}{2},m=\frac{1}{2},n=\frac{1}{\sqrt{2}} \end{gathered}$$
∴ Equation of line passing through (x₁, y₁, z₁) and direction cosines l, m, n is given by

$$\begin{gathered} \frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n} \\ \mathrm{i}.\mathrm{e}.\frac{x-2}{{\displaystyle \frac{1}{2}}}=\frac{y-1}{{\displaystyle \frac{1}{2}}}=\frac{z+1}{{\displaystyle \frac{1}{\sqrt{2}}}} \\ \mathrm{i}.\mathrm{e}.\frac{x-2}{1}=\frac{y-1}{1}=\frac{z+1}{\sqrt{2}}\left(\mathrm{multiplying}\mathrm{denominator}\mathrm{by}2\right) \end{gathered}$$