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

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo