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Question
Show that the line through the points (1, −1, 2) (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
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Solution
Let A(1, −1, 2), B(3, 4, −2), C(0, 3, 2), D(3, 5, 6)
Direction ratio of AB = (3 − 1, 4 − (−1), −2 − 2)
= (2, 5, −4)
Direction ratio of CD = (3 − 0, 5 − 3, 6 − 2)
= (3, 2, 4)
a1a2 + b1b2 + c1c2
= 2 × 3 + 5 × 2 + (−4) × 4
= 6 + 10 − 16
= 16 − 16
= 0
Hence, lines AB and CD are perpendicular.
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