Question

Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and, (1, 2, 5).

Answer 1

Equations of lines passing through the points $\left(x_1,y_1,z_1\right)\mathrm{and}\left(x_2,y_2,z_2\right)$are given by

$\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$

So, the equation of a line passing through (4, 7, 8) and (2, 3, 4) is

$$\begin{gathered} \frac{x-4}{2-4}=\frac{y-7}{3-7}=\frac{z-8}{4-8} \\ \Rightarrow \frac{x-4}{-2}=\frac{y-7}{-4}=\frac{z-8}{-4} \end{gathered}$$

Also, the equation of the line passing through the points ($-$1, $-$2, 1) and (1, 2, 5) is

$$\begin{gathered} \frac{x+1}{1+1}=\frac{y+2}{2+2}=\frac{z-1}{5-1} \\ \Rightarrow \frac{x+1}{2}=\frac{y+2}{4}=\frac{z-1}{4} \end{gathered}$$

We know that two lines are parallel if
$$\begin{gathered} \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2} \\ \mathrm{Cartesian}\mathrm{equations}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{lines}\mathrm{are}\mathrm{given}\mathrm{by} \\ \frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}\mathrm{and}\frac{\mathrm{x}-x_2}{a_2}=\frac{y-y_{\mathit{2}}}{b_2}=\frac{z-z_2}{c_2} \end{gathered}$$


We observe

$\frac{-2}{2}=\frac{-4}{4}=\frac{-4}{4}=-1$

Hence, the given lines are parallel to each other.