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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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.