
Miscellaneous
Misc 2
Misc 3 Deleted for CBSE Board 2022 Exams
Misc 4 Important
Misc 5 Important
Misc 6 Important
Misc 7
Misc 8 Important
Misc 9 Important
Misc 10
Misc 11 Important
Misc 12 Important
Misc 13 Important
Misc 14 Important
Misc 15 Important
Misc 16
Misc 17 Important
Misc 18 Important
Misc 19
Misc 20 Important
Misc 21 Important
Misc 22 (MCQ) Important
Misc 23 (MCQ) Important
Miscellaneous
Last updated at Dec. 11, 2021 by Teachoo
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