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Misc 1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to line determined by the points (3, 5, – 1), (4, 3, –1).Two lines having direction ratios 𝑎1, 𝑏1 , 𝑐1 and 𝑎2, 𝑏2, 𝑐2 are Perpendicular to each other if 𝒂1 𝒂2 + 𝒃1 𝒃2 + 𝒄1 𝒄2 = 0 Also, a line passing through (x1, y1, z1) and (x2, y2, z2) has the direction ratios (x2 − x1), (y2 − y1), (z2 − z1) We have two lines here: Line joining Origin O (0, 0, 0) and point A (2, 1, 1) Line joining points B (3, 5, -1) and C (4, 3, −1) Finding Direction ratios of both lines Line O (0, 0, 0) & A (2, 1, 1) Direction ratios : = (2 − 0), (1 − 0), (1 − 0) = 2, 1, 1 ∴ 𝒂1 = 2, 𝒃1 = 1, 𝒄1 = 1 Line B (3, 5, −1) & C (4, 3, −1) Direction ratios: = (4 − 3), (3 − 5), ( −1 + 1) = 1, −2, 0 ∴ 𝒂2 = 1, 𝒃2 = −2, 𝒄2 = 0 Now, 𝒂1 𝒂2 + 𝒃1 𝒃2 + 𝒄1 𝒄2 = (2 × 1) + (1 × −2) + (1 × 0) = 2 + (−2) + 0 = 2 − 2 = 0 Therefore, the given two lines are perpendicular

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.