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Question
Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (−1, −2, 1), (1, 2, 5).
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Solution
Let A(4, 7, 8), B(2, 3, 4), C(−1, −2, 1), D(1, 2, 5)
Direction ratio of AB = (2 − 4, 3 − 7, 4 − 8)
= (−2, −4, −4)
a1 = −2, b1 = −4, c1 = −4
Direction ratio of CD = (1 − (−1), 2 − (−2), 5 − 1)
= (2, 4, 4)
a2 = 2, b2 = 4, c2 = 4
`a_1/a_2 = (-2)/2 = -1`
`b_1/b_2 = (-4)/4 = -1`
`c_1/c_2 = (-4)/4 = -1`
Hence, `a_1/a_2 = b_1/b_2 = c_1/c_2`
Lines AB and CD are parallel.
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