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Example 21 - Show that points A, B, C are collinear - Chapter 10 - Collinearity of 3 points or 3 position vectors

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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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﷯﷯ = ﷮ 3﷮2﷯+ −1﷯﷮2﷯+ −2﷯﷮2﷯﷯ 𝐴𝐵﷯﷯ = ﷮9+1+4﷯ = ﷮𝟏𝟒﷯ Magnitude of 𝐵𝐶﷯﷯ = ﷮ 6﷮2﷯+ −2﷯﷮2﷯+ −4﷯﷮2﷯﷯ 𝐵𝐶﷯﷯ = ﷮36+4+16﷯ = ﷮56﷯ = ﷮4×14﷯ = 2 ﷮𝟏𝟒﷯ Magnitude of 𝐴𝐶﷯﷯ = ﷮ 9﷮2﷯+ −3﷯﷮2﷯+ −6﷯﷮2﷯﷯ 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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