Supplementary Exercise Q7
Last updated at Aug. 24, 2021 by Teachoo
Last updated at Aug. 24, 2021 by Teachoo
Transcript
Supplementary Exercise Q7 Show that the four points A, B, C and D with position vectors 4๐ ฬ + 5๐ ฬ + ๐ ฬ, โ(๐ ฬ + ๐ ฬ), 3๐ ฬ + 9๐ ฬ + 4๐ ฬ and โ4๐ ฬ + 4๐ ฬ + 4๐ ฬ, respectively are co-planar 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 + 5๐ ฬ + ๐ ฬ) B (โ๐ ฬ โ ๐ ฬ) (๐จ๐ฉ) โ = (0๐ ฬ โ ๐ ฬ โ ๐ ฬ) โ (4๐ ฬ + 5๐ ฬ + ๐ ฬ) = โ4๐ ฬ โ 6๐ ฬ โ 2๐ ฬ A (4๐ ฬ + 5๐ ฬ + ๐ ฬ) C (3๐ ฬ + 9๐ ฬ + 4๐ ฬ) (๐จ๐ช) โ = (3๐ ฬ + 9๐ ฬ + 4๐ ฬ) โ (4๐ ฬ + 5๐ ฬ + ๐ ฬ) = โ๐ ฬ + 4๐ ฬ + 3๐ ฬ A (4๐ ฬ + 5๐ ฬ + ๐ ฬ) D (โ4๐ ฬ + 4๐ ฬ + 4๐ ฬ) (๐จ๐ซ) โ = (โ4๐ ฬ + 4๐ ฬ + 4๐ ฬ) โ (4๐ ฬ + 5๐ ฬ + ๐ ฬ) = โ8๐ ฬ โ ๐ ฬ + 3๐ ฬ [(๐ด๐ต) โ, (๐ด๐ถ) โ, (๐ด๐ท) โ ] = |โ 8(โ4&โ6&โ2@โ1&4&3@โ8&โ1&3)| = โ4[(4ร3)โ(โ1ร3) ] โ (โ6) [(โ1 ร 3) โ (โ8 ร 3)] + (โ2)[(โ1รโ1)โ(โ8ร4) ] = โ4 [12+3]+6[โ3+24]โ2[1+32] = โ4 (15) + 6 (21) โ 2 (33) = โ60 + 126 โ 66 = โ126+ 126 = 0 โด[(๐ด๐ต) โ, (๐ด๐ถ) โ, (๐ด๐ท) โ ] = 0 Therefore, points A, B, C and D are coplanar.
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