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

Made by

Davneet Singh

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

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