Misc 1 - Chapter 11 Class 12 Three Dimensional Geometry (Term 2)

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Misc 1 Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the 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) A (0, 0, 0) B (2, 1, 1) Direction ratios : (2 − 0), (1 − 0), (1 − 0) = 2, 1, 1 ∴ 𝒂1 = 2, 𝒃1 = 1, 𝒄1 = 1 C (3, 5, −1) D (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