Last updated at Dec. 24, 2019 by Teachoo

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Misc 8(Introduction) Show that the points A(1, â 2, â 8), B (5, 0, â 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC. (1) Three points collinear i.e. AB + BC = AC (2) Three vectors. i.e. đ´đĩīˇ¯īˇ¯ + đĩđļīˇ¯īˇ¯ = đ´đļīˇ¯īˇ¯ Misc 8 Show that the points A(1, â 2, â 8), B (5, 0, â 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC. 3 points A, B, C are collinear if i.e. đ¨đŠīˇ¯īˇ¯ + đŠđĒīˇ¯īˇ¯ = đ¨đĒīˇ¯īˇ¯ A (1, -2, â8) B (5, 0,â2) C (11, 3, 7) đ´đĩīˇ¯ = (5 â 1) đīˇ¯ + (0 â (â2)) đīˇ¯ + (â2â(â8)) đīˇ¯ = 4 đīˇ¯ + 2 đīˇ¯ + 6 đīˇ¯ đĩđļīˇ¯ = (11 â 5) đīˇ¯ + (3 â 0) đīˇ¯ + (7â(â2)) đīˇ¯ = 6 đīˇ¯ + 3 đīˇ¯ + 9 đīˇ¯ đ´đļīˇ¯ = (11 â 1) đīˇ¯ + (3 â (â2)) đīˇ¯ + (7â(â8)) đīˇ¯ = 10 đīˇ¯ + 5 đīˇ¯ + 15 đīˇ¯ Magnitude of đ´đĩīˇ¯ = īˇŽ42+22+62īˇ¯ đ´đĩīˇ¯ = īˇŽ16+4+36īˇ¯ = īˇŽ56īˇ¯ = īˇŽ4Ã14 īˇ¯ = 2 īˇŽ14īˇ¯ Magnitude of đĩđļīˇ¯ = īˇŽ62+32+92īˇ¯ đĩđļīˇ¯īˇ¯= īˇŽ36+9+81īˇ¯ = īˇŽ126īˇ¯ = īˇŽ9Ã14 īˇ¯ = 3 īˇŽ14īˇ¯ Magnitude of đ´đļīˇ¯ = īˇŽ102+52+152īˇ¯ đ´đļīˇ¯īˇ¯= īˇŽ100+25+225īˇ¯= īˇŽ350īˇ¯ = īˇŽ25Ã14 īˇ¯ = 5 īˇŽ14īˇ¯ đ´đĩīˇ¯īˇ¯ + đĩđļīˇ¯īˇ¯ = 2 īˇŽ14 īˇ¯ + 3 īˇŽ14 īˇ¯ = 5 īˇŽ14 īˇ¯ = đ´đļīˇ¯īˇ¯ â´ A, B and C are collinear. Finding the ratio in which B divides AC Let B divide AC in the ratio k : 1 đđ´īˇ¯ = 1 đīˇ¯ â 2 đīˇ¯ â 8 đīˇ¯ đđĩīˇ¯ = 5 đīˇ¯ + 0 đīˇ¯ â 2 đīˇ¯ and đđļīˇ¯ = 11 đīˇ¯ + 3 đīˇ¯ + 7 đīˇ¯ Position vector of đĩ = đ đđļīˇ¯+1. đđ´īˇ¯īˇŽđ+1īˇ¯ đđĩīˇ¯ = đ 11 đīˇ¯ + 3 đīˇ¯ + 7 đīˇ¯īˇ¯ + 1 1 đīˇ¯ â2 đīˇ¯ â 8 đīˇ¯īˇ¯īˇŽđ + 1īˇ¯ 5 đīˇ¯ + 0 đīˇ¯ â 2 đīˇ¯ = 11đ đīˇ¯ + 3đ đīˇ¯ + 7đ đīˇ¯ + đīˇ¯ â 2 đīˇ¯ â 8 đīˇ¯īˇŽđ + 1īˇ¯ 5 đīˇ¯ + 0 đīˇ¯ â 2 đīˇ¯ = 11đ + 1īˇ¯ đīˇ¯ + 3đ â 2īˇ¯ đīˇ¯ + (7đâ 8) đīˇ¯īˇŽđ + 1īˇ¯ â´ 5 đīˇ¯ + 0 đīˇ¯ â 2 đīˇ¯ = (11đ + 1) īˇŽ(đ + 1)īˇ¯ đīˇ¯ + (3đ â 2) īˇŽ(đ + 1)īˇ¯ đīˇ¯ + (7đ â 8) īˇŽ(đ + 1)īˇ¯ đīˇ¯ Since the two vectors are equal, corresponding components are also equal. So, 11đ+1īˇŽđ+1īˇ¯ = 5 11k + 1 = 5k + 5 11k â 5k = 5 â 1 6k = 4 k = 4īˇŽ6īˇ¯ = 2īˇŽ3īˇ¯ Thus, B divides AC in the ratio 2īˇŽ3īˇ¯ : 1 or 2 : 3

Miscellaneous

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Misc 2

Misc 3 Important

Misc 4

Misc 5 Important

Misc 6

Misc 7 Important

Misc 8 Important You are here

Misc 9

Misc 10 Important

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.