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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).
Solution
Equations of lines passing through the points
\[\left( x_1 , y_1 , z_1 \right) \text { and } \left ( x_2 , y_2 , z_2 \right)\] 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
\[\frac{x - 4}{2 - 4} = \frac{y - 7}{3 - 7} = \frac{z - 8}{4 - 8}\]
\[ \Rightarrow \frac{x - 4}{- 2} = \frac{y - 7}{- 4} = \frac{z - 8}{- 4}\]
Also, the equation of the line passing through the points ( -1,-2 ,1) and (1,2,5) is
\[\frac{x + 1}{1 + 1} = \frac{y + 2}{2 + 2} = \frac{z - 1}{5 - 1}\]
\[ \Rightarrow \frac{x + 1}{2} = \frac{y + 2}{4} = \frac{z - 1}{4}\]
We know that two lines are parallel if
\[\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
\[\text { Cartesian equations of the two lines are given by } \]
\[\frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1} \text { and } \frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2}\]
We observe
\[\frac{- 2}{2} = \frac{- 4}{4} = \frac{- 4}{4} = - 1\]
Hence, the given lines are parallel to each other.
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