Question

Cartesian equations of a line AB are $\frac{2x-1}{2}=\frac{4-y}{7}=\frac{z+1}{2}.$ Write the direction ratios of a line parallel to AB.

Answer 1

We have

$\frac{2x-1}{2}=\frac{4-y}{7}=\frac{z+1}{2}$

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

$\frac{x-{\displaystyle \frac{1}{2}}}{1}=\frac{y-4}{-7}=\frac{z+1}{2}$

The direction ratios of the line parallel to AB are proportional to 1, $-$7, 2.

Also, the direction cosines of the line parallel to AB are proportional to

$$\begin{gathered} \frac{1}{\sqrt{1^2+{\left(-7\right)}^2+2^2}},\frac{-7}{\sqrt{1^2+{\left(-7\right)}^2+2^2}},\frac{2}{\sqrt{1^2+{\left(-7\right)}^2+2^2}} \\ =\frac{1}{\sqrt{54}},\frac{-7}{\sqrt{54}},\frac{2}{\sqrt{54}} \end{gathered}$$