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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

We know that the direction ratios of the line passing through the points x1, y1, z1 and x2, y2, z2 are x2-x1, y2-y1, z2-z1.Let the first two points be A 4, 7, 8 and B 2, 3, 4.Thus, the direction ratios of AB are 2-4, 3-7, 4-8, i.e. -2, -4, -4.Similarly, let the other two points be C -1,-2, 1 and D1, 2, 5.Thus, the direction ratios of CD are 1--1, 2--2, 5-1, i.e. 2, 4, 4.It can be seen that the direction ratios of CD are-1 times that of AB, i.e. they are proportional.Therefore, AB and CD are parallel lines.

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