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Ex 10.5, 7 (Supplementary NCERT) - Show that a, b, c are coplanar if

Ex 10.5 , 7 (Supplementary NCERT) - Chapter 10 Class 12 Vector Algebra - Part 2
Ex 10.5 , 7 (Supplementary NCERT) - Chapter 10 Class 12 Vector Algebra - Part 3

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Ex 10.5, 7 (Supplementary NCERT) Show that vectors 𝑎 ⃗, 𝑏 ⃗ and 𝑐 ⃗ are coplanar if 𝑎 ⃗ + 𝑏 ⃗, 𝑏 ⃗ + 𝑐 and 𝑐 ⃗ + 𝑎 ⃗ are coplanarGiven 𝑎 ⃗ + 𝑏 ⃗, 𝑏 ⃗ + 𝑐 and 𝑐 ⃗ + 𝑎 ⃗ are coplanar [■8(𝒂 ⃗" + " 𝒃 ⃗&𝒃 ⃗+𝒄 ⃗&𝒄 ⃗+𝒂 ⃗ )] = 0 We need to prove 𝑎 ⃗, 𝑏 ⃗ and 𝑐 ⃗ are coplanar i.e. [■8(𝒂 ⃗&𝒃 ⃗&𝒄 ⃗ )] = 0 Now, [■8(𝑎 ⃗" + " 𝑏 ⃗&𝑏 ⃗+𝑐 ⃗&𝑐 ⃗+𝑎 ⃗ )] = 0 (𝒂 ⃗" + " 𝒃 ⃗). ["(" 𝒃 ⃗+𝒄 ⃗") " × " (" 𝒄 ⃗+𝒂 ⃗)] = 0 (𝑎 ⃗" + " 𝑏 ⃗). ["(" 𝑏 ⃗ × 𝑐 ⃗") + (" 𝑏 ⃗ × 𝑎 ⃗)+"(" 𝒄 ⃗ × 𝒄 ⃗") + (" 𝑐 ⃗ × 𝑎 ⃗)] = 0 "(" 𝑎 ⃗+𝑏 ⃗")." [(𝑏 ⃗ × 𝑐 ⃗") + (" 𝑏 ⃗ × 𝑎 ⃗ ) "+ 0 + (" 𝑐 ⃗" × " 𝑎 ⃗") " ] = 0 𝑎 ⃗. (𝑏 ⃗ × 𝑐 ⃗) + 𝑏 ⃗.(𝑏 ⃗ × 𝑐 ⃗) + 𝑎 ⃗. (𝑏 ⃗ × 𝑎 ⃗) + 𝑏 ⃗.(𝑏 ⃗ × 𝑎 ⃗) + 𝑎 ⃗.(𝑐 ⃗ × 𝑎 ⃗) + 𝑏 ⃗.(𝑐 ⃗ × 𝑎 ⃗) = 0 [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + [𝒃 ⃗", " 𝒃 ⃗", " 𝒄 ⃗ ] + [𝒂 ⃗", " 𝒃 ⃗", " 𝒂 ⃗ ] + [𝒃 ⃗", " 𝒃 ⃗", " 𝒂 ⃗ ] + [𝒂 ⃗", " 𝒄 ⃗", " 𝒂 ⃗ ] + [𝑏 ⃗", " 𝑐 ⃗", " 𝑎 ⃗ ] = 0 [𝒃 ⃗", " 𝒃 ⃗", " 𝒄 ⃗ ] = [𝑐 ⃗", " 𝑏 ⃗", " 𝑏 ⃗ ] = 𝑐 ⃗. (𝑏 ⃗ × 𝑏 ⃗) As (𝑏 ⃗ × 𝑏 ⃗) = 0 ⃗ = 𝑐 ⃗ . 0 ⃗ = 0 Using Prop: [𝒃 ⃗", " 𝒃 ⃗", " 𝒄 ⃗ ] = 𝟎 [𝒂 ⃗", " 𝒃 ⃗", " 𝒂 ⃗ ] = 𝟎 [𝒃 ⃗", " 𝒃 ⃗", " 𝒂 ⃗ ] = 𝟎 [𝒂 ⃗", " 𝒄 ⃗", " 𝒂 ⃗ ] = 𝟎 [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + 0 + 0 + 0 + 0 + [𝑏 ⃗", " 𝑐 ⃗", " 𝑎 ⃗ ] = 0 [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + [𝒃 ⃗", " 𝒄 ⃗", " 𝒂 ⃗ ] = 0 [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] + [𝒂 ⃗", " 𝒃 ⃗", " 𝒄 ⃗ ] = 0 2[𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] = 0 [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] = 0 Since [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] = 0, 𝑎 ⃗, 𝑏 ⃗ and 𝑐 ⃗ are coplanar As [𝑎 ⃗", " 𝑏 ⃗", " 𝑐 ⃗ ] = [𝑐 ⃗", " 𝑎 ⃗", " 𝑏 ⃗ ]= [𝑏 ⃗", " 𝑐 ⃗", " 𝑎 ⃗ ]

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.