Last updated at March 11, 2017 by Teachoo

Transcript

Example 21 (Introduction) Show that the points A(−2 𝑖 + 3 𝑗 + 5 𝑘), B( 𝑖 + 2 𝑗 + 3 𝑘) and C(7 𝑖 − 𝑘) are collinear. (1) Three points collinear i.e. AB + BC = AC (2) Three vectors collinear i.e. 𝐴𝐵 + 𝐵𝐶 = 𝐴𝐶 Example 21 Show that the points A(−2 𝑖 + 3 𝑗 + 5 𝑘), B( 𝑖 + 2 𝑗 + 3 𝑘) and C(7 𝑖 − 𝑘) are collinear. 3 points A, B, C are collinear if 𝑨𝑩 + 𝑩𝑪 = 𝑨𝑪 A (−2 𝑖 + 3 𝑗 + 5 𝑘) B (1 𝑖 + 2 𝑗 + 3 𝑘) C (7 𝑖 + 0 𝑗 − 1 𝑘) 𝐴𝐵 = (1 – (-2)) 𝑖 + (2 − 3) 𝑗 + (3 − 5) 𝑘 = 3 𝑖 – 1 𝑗 – 2 𝑘 𝐵𝐶 = (7 − 1) 𝑖 + (0 − 2) 𝑗 + (-1−3) 𝑘 = 6 𝑖 – 2 𝑗 – 4 𝑘 𝐴𝐶 = (7 − (-2)) 𝑖 + (0 − 3) 𝑗 + (-1 − 5) 𝑘 = 9 𝑖 – 3 𝑗 – 6 𝑘 Magnitude of 𝐴𝐵 = 32+ −12+ −22 𝐴𝐵 = 9+1+4 = 𝟏𝟒 Magnitude of 𝐵𝐶 = 62+ −22+ −42 𝐵𝐶 = 36+4+16 = 56 = 4×14 = 2 𝟏𝟒 Magnitude of 𝐴𝐶 = 92+ −32+ −62 𝐴𝐶 = 81+9+36 = 126 = 9×14 = 3 𝟏𝟒 Thus, 𝐴𝐵 + 𝐵𝐶 = 14 + 2 14 = 3 14 = 𝐴𝐶 Thus, A, B and C are collinear.

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.