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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’[email protected]βˆ’1&4&[email protected]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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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.