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Question
Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).
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Solution
\[\text { We know that the direction ratios of the line passing through the points } \left( x_1 , y_1 , z_1 \right) \text { and } \left( x_2 , y_2 , z_2 \right) \text { are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]
\[\text{ Let the first two points be A } \left( 4, 7, 8 \right) \text{ and } B \left( 2, 3, 4 \right) . \]
\[\text{ Thus, the direction ratios of AB are } \left( 2 - 4 \right), \left( 3 - 7 \right), \left( 4 - 8 \right), \text{ i . e } . - 2, - 4, - 4 . \]
\[\text{ Similarly, let the other two points be C } \left( - 1, - 2, 1 \right) \text{ and } D\left( 1, 2, 5 \right) . \]
\[\text{ Thus, the direction ratios of CD are } \left[ 1 - \left( - 1 \right) \right], \left[ 2 - \left( - 2 \right) \right], \left( 5 - 1 \right),\text{ i . e} . 2, 4, 4 . \]
\[\text{ It can be seen that the direction ratios of CD are - 1 times that of AB, i . e . they are proportional }. \]
\[\text { Therefore, AB and CD are parallel lines } .\]
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