Supplementary Example 5
Last updated at Aug. 24, 2021 by
Last updated at Aug. 24, 2021 by
Transcript
Supplementary Example 5 Show that the four points with position vectors 6๐ ฬ โ 7๐ ฬ, 16๐ ฬ โ 19๐ ฬ โ 4๐ ฬ, 3๐ ฬ โ 6๐ ฬ & 2๐ ฬ + 5๐ ฬ + 10๐ ฬ are not co-planar Let points be A = 6๐ ฬ โ 7๐ ฬ B = 16๐ ฬ โ 19๐ ฬ โ 4๐ ฬ C = 3๐ ฬ โ 6๐ ฬ D = 2๐ ฬ + 5๐ ฬ +10๐ ฬ Four points A, B, C, D are coplanar if the three vectors (๐ด๐ต) โ , (๐ด๐ถ) โ and (๐ด๐ท) โ are coplanar. i.e. [(๐จ๐ฉ) โ, (๐จ๐ช) โ, (๐จ๐ซ) โ ] = 0 A (6๐ ฬ โ 7๐ ฬ) B (16๐ ฬ โ 19๐ ฬ โ 4๐ ฬ) (๐จ๐ฉ) โ = (16๐ ฬ โ 19๐ ฬ โ 4๐ ฬ) โ (6๐ ฬ โ 7๐ ฬ + 0๐ ฬ) = (16 โ 6) ๐ ฬ + (โ19 + 7) ๐ ฬ โ 4๐ ฬ = 10๐ ฬ โ 12๐ ฬ โ 4๐ ฬ A (6๐ ฬ โ 7๐ ฬ) C (3๐ ฬ โ 6๐ ฬ) (๐จ๐ช) โ = (0๐ ฬ + 3๐ ฬ โ 6๐ ฬ) โ (6๐ ฬ โ 7๐ ฬ + 0๐ ฬ) = (0 โ 6) ๐ ฬ + (3 + 7) ๐ ฬ + (โ6 โ 0) ๐ ฬ = โ6๐ ฬ + 10๐ ฬ โ 6๐ ฬ A (6๐ ฬ โ 7๐ ฬ) D (2๐ ฬ + 5๐ ฬ +10๐ ฬ) (๐จ๐ซ) โ = (2๐ ฬ + 5๐ ฬ +10๐ ฬ) โ (6๐ ฬ โ 7๐ ฬ + 0๐ ฬ) = (2 โ 6) ๐ ฬ + (5 + 7) ๐ ฬ + (10 โ 0) ๐ ฬ = โ4๐ ฬ + 12๐ ฬ + 10๐ ฬ Now, [(๐ด๐ต) โ, (๐ด๐ถ) โ, (๐ด๐ท) โ ] = |โ 8(10&โ12&โ4@โ6&10&โ6@โ4&12&10)| = 10[(10ร10)โ(12รโ6) ] โ (โ12) [(โ6ร10)โ(โ4รโ6)] + (โ4)[(โ6ร12)โ(โ4ร10) ] = 10[100+72]+12[โ60โ24]โ4[โ72+40] = 10[172]+12[โ84]โ4[โ32] = 1720 โ 1008 + 128 = 840 โด [(๐จ๐ฉ) โ, (๐จ๐ช) โ, (๐จ๐ซ) โ ] โ 0 Therefore, points A, B, C and D are not coplanar.
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