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Example 29 (Supplementary NCERT) - Show that A, B, C, D with position

Example 29 (Supplementary NCERT) - Chapter 10 Class 12 Vector Algebra - Part 2

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Example 29 (Supplementary NCERT) Show that the four points A, B, C and D with position vectors 4𝑖 Μ‚ + 5𝑗 Μ‚ + π‘˜ Μ‚, βˆ’(𝑗 Μ‚ + π‘˜ Μ‚), 3𝑖 Μ‚ + 9𝑗 Μ‚ + 4π‘˜ Μ‚ & βˆ’4𝑖 Μ‚ + 4𝑗 Μ‚ + 4π‘˜ Μ‚, respectively coplanar Four points A, B, C, D are coplanar if the three vectors (𝐴𝐡) βƒ— , (𝐴𝐢) βƒ— and (𝐴𝐷) βƒ— are coplanar. i.e. [(𝑨𝑩) βƒ—, (𝑨π‘ͺ) βƒ—, (𝑨𝑫) βƒ— ] = 0 A (4𝑖 Μ‚ + 5𝑗 Μ‚ + π‘˜ Μ‚) B (βˆ’π‘— Μ‚ βˆ’ π‘˜ Μ‚) (𝑨𝑩) βƒ— = (0𝑖 Μ‚ βˆ’ 𝑗 Μ‚ βˆ’ π‘˜ Μ‚) βˆ’ (4𝑖 Μ‚ + 5𝑗 Μ‚ + π‘˜ Μ‚) = βˆ’4π’Š Μ‚ βˆ’ 6𝒋 Μ‚ βˆ’ 2π’Œ Μ‚ A (4𝑖 Μ‚ + 5𝑗 Μ‚ + π‘˜ Μ‚) C (3𝑖 Μ‚ + 9𝑗 Μ‚ + 4π‘˜ Μ‚) (𝑨π‘ͺ) βƒ— = (3𝑖 Μ‚ + 9𝑗 Μ‚ + 4π‘˜ Μ‚) βˆ’ (4𝑖 Μ‚ + 5𝑗 Μ‚ + π‘˜ Μ‚) = β€“π’Š Μ‚ + 4𝒋 Μ‚ + 3π’Œ Μ‚ A (4𝑖 Μ‚ + 5𝑗 Μ‚ + π‘˜ Μ‚) D (βˆ’4𝑖 Μ‚ + 4𝑗 Μ‚ + 4π‘˜ Μ‚) (𝑨𝑫) βƒ— = (βˆ’4𝑖 Μ‚ + 4𝑗 Μ‚ + 4π‘˜ Μ‚) βˆ’ (4𝑖 Μ‚ + 5𝑗 Μ‚ + π‘˜ Μ‚) = –8π’Š Μ‚ βˆ’ 𝒋 Μ‚ + 3π’Œ Μ‚ [(𝐴𝐡) βƒ—, (𝐴𝐢) βƒ—, (𝐴𝐷) βƒ— ] = |β– 8(βˆ’4&βˆ’6&βˆ’2@βˆ’1&4&3@βˆ’8&βˆ’1&3)| = βˆ’4[(4Γ—3)βˆ’(βˆ’1Γ—3) ] βˆ’ (βˆ’6) [(βˆ’1Γ—3)βˆ’(βˆ’8Γ—3)] + (βˆ’2)[(βˆ’1Γ—βˆ’1)βˆ’(βˆ’8Γ—4) ] = –4 [12+3]+6[βˆ’3+24]βˆ’2[1+32] = βˆ’4 (15) + 6 (21) βˆ’ 2 (33) = βˆ’60 + 126 βˆ’ 66 = βˆ’126+ 126 = 0 ∴[(𝐴𝐡) βƒ—, (𝐴𝐢) βƒ—, (𝐴𝐷) βƒ— ] = 0 Therefore, points A, B, C and D are coplanar.

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