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Last updated at Sept. 17, 2018 by Teachoo
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Ex 10.5, 5 (Supplementary NCERT) Show that the four points with position vectors 4๐ ฬ + 8๐ ฬ + 12๐ ฬ, 2๐ ฬ + 4๐ ฬ + 6๐ ฬ, 3๐ ฬ + 5๐ ฬ + 4๐ ฬ and 5๐ ฬ + 8๐ ฬ + 5๐ ฬ are coplanarLet points be A = 4๐ ฬ + 8๐ ฬ + 12๐ ฬ B = 2๐ ฬ + 4๐ ฬ + 6๐ ฬ C = 3๐ ฬ + 5๐ ฬ + 4๐ ฬ D = 5๐ ฬ + 8๐ ฬ + 5๐ 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 (4๐ ฬ + 8๐ ฬ + 12๐ ฬ) B (2๐ ฬ + 4๐ ฬ + 6๐ ฬ) (๐จ๐ฉ) โ = (2๐ ฬ + 4๐ ฬ + 6๐ ฬ) โ (4๐ ฬ + 8๐ ฬ + 12๐ ฬ) = (2 โ 4) ๐ ฬ + (4 โ 8) ๐ ฬ + (6 โ 12)๐ ฬ = โ2๐ ฬ โ 4๐ ฬ โ 6๐ ฬ A (4๐ ฬ + 8๐ ฬ + 12๐ ฬ) C (3๐ ฬ + 5๐ ฬ + 4๐ ฬ) (๐จ๐ช) โ = (3๐ ฬ + 5๐ ฬ + 4๐ ฬ) โ (4๐ ฬ + 8๐ ฬ + 12๐ ฬ) = (3 โ 4) ๐ ฬ + (5 โ 8) ๐ ฬ + (4 โ 12) ๐ ฬ = โ๐ ฬ โ 3๐ ฬ โ 8๐ ฬ A (4๐ ฬ + 8๐ ฬ + 12๐ ฬ) D (5๐ ฬ + 8๐ ฬ + 5๐ ฬ) (๐จ๐ซ) โ = (5๐ ฬ + 8๐ ฬ + 5๐ ฬ) โ (4๐ ฬ + 8๐ ฬ + 12๐ ฬ) = (5 โ 4) ๐ ฬ + (8 โ 8) ๐ ฬ + (5 โ 12) ๐ ฬ = ๐ ฬ + 0๐ ฬ โ 7๐ ฬ Now, [(๐ด๐ต) โ, (๐ด๐ถ) โ, (๐ด๐ท) โ ] = |โ 8(โ2&โ4&โ6@โ1&โ3&โ8@1&0&โ7)| = โ2[(โ3รโ7)โ(0รโ8) ] โ (โ4) [(โ1รโ7)โ(1รโ8)] + (โ6)[(โ1ร0)โ(1รโ3) ] = โ2[21โ0]+4[7+8]โ6[0+3] = โ2[21]+4[15]โ6[3] = โ 42 + 60 โ 18 = 60 โ 60 = 0 โด [(๐จ๐ฉ) โ, (๐จ๐ช) โ, (๐จ๐ซ) โ ] = 0 Therefore, points A, B, C and D are coplanar.
Ex 10.5 (Supplementary NCERT)
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