Question

Write the direction cosines of the line $\frac{x-2}{2}=\frac{2y-5}{-3},z=2.$

Answer 1

We have

$\frac{x-2}{2}=\frac{2y-5}{-3},z=2$

The equation of the given line can be re-written as

$$\begin{gathered} \frac{x-2}{2}=\frac{y-{\displaystyle \frac{5}{2}}}{-{\displaystyle \frac{3}{2}}}=\frac{z-2}{0} \\ \Rightarrow \frac{x-2}{4}=\frac{y-{\displaystyle \frac{5}{2}}}{-3}=\frac{z-2}{0} \end{gathered}$$


The direction ratios of the given line are proportional to 4, $-$3, 0.

Hence, the direction cosines of the given line are proportional to

$$\begin{gathered} \frac{4}{\sqrt{4^2+{\left(-3\right)}^2+0^2}},\frac{-3}{\sqrt{4^2+{\left(-3\right)}^2+0^2}},\frac{0}{\sqrt{4^2+{\left(-3\right)}^2+0^2}} \\ =\frac{4}{5},\frac{-3}{5},0 \end{gathered}$$