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If a, b, c are three vectors such that a + b + c = 0 ,Β  then prove that a Γ— b = b Γ— c = c Γ— a, and hence show that [a b c] = 0.

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

You can download the question paper hereΒ  https://www.teachoo.com/cbse/sample-papers/


If a, b, c are three vectors such that a + b + c = 0, then prove a x b

Question 20 - CBSE Class 12 Sample Paper for 2018 Boards - Part 2
Question 20 - CBSE Class 12 Sample Paper for 2018 Boards - Part 3 Question 20 - CBSE Class 12 Sample Paper for 2018 Boards - Part 4

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Transcript

Question 20 If π‘Ž βƒ—, 𝑏 βƒ—, 𝑐 βƒ— are three vectors such that π‘Ž βƒ— + 𝑏 βƒ— + 𝑐 βƒ— = 0 βƒ— , then prove that π‘Ž βƒ— Γ— 𝑏 βƒ— = 𝑏 βƒ— Γ— 𝑐 βƒ— = 𝑐 βƒ— Γ— π‘Ž βƒ—, and hence show that [π‘Ž βƒ—" " 𝑏 βƒ—" " 𝑐 βƒ— ] = 0. Theory Here [π‘Ž βƒ—" " 𝑏 βƒ—" " 𝑐 βƒ— ] = π‘Ž βƒ—.(𝑏 βƒ— Γ— 𝑐 βƒ— ) Given π‘Ž βƒ— + 𝑏 βƒ— + 𝑐 βƒ— = 0 βƒ— π‘Ž βƒ—Γ—(π‘Ž βƒ—+𝑏 βƒ—+𝑐 βƒ— )= π‘Ž βƒ—Γ—0 βƒ— π‘Ž βƒ—Γ—π‘Ž βƒ—+π‘Ž ⃗×𝑏 βƒ—+π‘Ž ⃗×𝑐 βƒ—= 0 βƒ— Since π‘Ž βƒ—Γ—π‘Ž βƒ—=0 " " 0+π‘Ž ⃗×𝑏 βƒ—+π‘Ž ⃗×𝑐 βƒ—=" " 0 βƒ— π‘Ž ⃗×𝑏 βƒ—+π‘Ž ⃗×𝑐 βƒ—=" " 0 βƒ— π‘Ž ⃗×𝑏 βƒ—=βˆ’π‘Ž ⃗×𝑐 βƒ— Since βˆ’π‘Ž ⃗×𝑐 βƒ— = 𝑐 βƒ—Γ—π‘Ž βƒ— 𝒂 ⃗×𝒃 βƒ—=𝒄 ⃗×𝒂 βƒ— Similarly, π‘Ž βƒ— + 𝑏 βƒ— + 𝑐 βƒ— = 0 βƒ— 𝑏 βƒ—Γ—(π‘Ž βƒ—+𝑏 βƒ—+𝑐 βƒ— )= 𝑏 βƒ—Γ—0 βƒ— 𝑏 βƒ—Γ—π‘Ž βƒ—+𝑏 ⃗×𝑏 βƒ—+𝑏 ⃗×𝑐 βƒ—= 0 βƒ— Since 𝑏 ⃗×𝑏 βƒ—=0 𝑏 βƒ—Γ—π‘Ž βƒ—+0+𝑏 ⃗×𝑐 βƒ—= 0 βƒ— 𝑏 βƒ—Γ—π‘Ž βƒ—+𝑏 ⃗×𝑐 βƒ—=" " 0 βƒ— 𝑏 ⃗×𝑐 βƒ—=βˆ’π‘ βƒ—Γ—π‘Ž βƒ— 𝑏 ⃗×𝑐 βƒ—=βˆ’π‘ βƒ—Γ—π‘Ž βƒ— Since βˆ’π‘ βƒ—Γ—π‘Ž βƒ— = π‘Ž ⃗×𝑏 βƒ— 𝑏 ⃗×𝑐 βƒ—=π‘Ž ⃗×𝑏 βƒ— Thus, 𝒂 ⃗×𝒃 βƒ—=𝒄 ⃗×𝒂 βƒ— & 𝑏 ⃗×𝑐 βƒ—=π‘Ž ⃗×𝑏 βƒ— ∴ 𝒂 ⃗×𝒃 βƒ—=𝒃 ⃗×𝒄 βƒ—=𝒄 ⃗×𝒂 βƒ— Now, we need to show that show that [π‘Ž βƒ—" " 𝑏 βƒ—" " 𝑐 βƒ— ] = 0 [π‘Ž βƒ— 𝑏 βƒ— 𝑐 βƒ— ]=π‘Ž βƒ— . (𝑏 ⃗×𝑐 βƒ— ) From (1): 𝑏 ⃗×𝑐 βƒ— = π‘Ž ⃗×𝑏 βƒ— =π‘Ž βƒ— . (π‘Ž ⃗×𝑏 βƒ— ) Now, π‘Ž ⃗×𝑏 βƒ— will be a vector perpendicular to π‘Ž βƒ— And dot product of π‘Ž βƒ— with a vector perpendicular to π‘Ž βƒ— will be 0 as angle is 90Β° and cos 90Β° = 0 ∴ [π‘Ž βƒ— 𝑏 βƒ— 𝑐 βƒ— ]=π‘Ž βƒ— . (π‘Ž ⃗×𝑏 βƒ— ) = 0 Hence proved

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.