Supplementary Example 6

Transcript

Supplementary Example 6 For any three vectors ∴ 𝑎﷯, 𝑏﷯, and 𝑐﷯, prove that 𝑎﷯ + 𝑏﷯﷮ 𝑏﷯+ 𝑐﷯﷮ 𝑐﷯+ 𝑎﷯﷯﷯ = 2 [ 𝑎﷯ , 𝑏﷯, 𝑐﷯] To prove 𝑎﷯ + 𝑏﷯﷮ 𝑏﷯+ 𝑐﷯﷮ 𝑐﷯+ 𝑎﷯﷯﷯ = 2 [ 𝑎﷯ , 𝑏﷯, 𝑐﷯] Solving LHS 𝑎﷯ + 𝑏﷯﷮ 𝑏﷯+ 𝑐﷯﷮ 𝑐﷯+ 𝑎﷯﷯﷯ = ( 𝑎﷯ + 𝑏﷯). ( 𝑏﷯+ 𝑐﷯) × ( 𝑐﷯+ 𝑎﷯)﷯ = ( 𝑎﷯ + 𝑏﷯). ( 𝑏﷯× 𝑐﷯) + ( 𝑏﷯× 𝑎﷯)+( 𝒄﷯× 𝒄﷯) + ( 𝑐﷯× 𝑎﷯)﷯ = ( 𝑎﷯+ 𝑏﷯). 𝑏﷯× 𝑐﷯) + ( 𝑏﷯× 𝑎﷯﷯ + 0 + ( 𝑐﷯ × 𝑎﷯) ﷯ = 𝑎﷯. ( 𝑏﷯ × 𝑐﷯) + 𝑏﷯.( 𝑏﷯ × 𝑐﷯) + 𝑎﷯. ( 𝑏﷯ × 𝑎﷯) + 𝑏﷯.( 𝑏﷯ × 𝑎﷯) + 𝑎﷯.( 𝑐﷯ × 𝑎﷯) + 𝑏﷯.( 𝑐﷯ × 𝑎﷯) = 𝑎﷯, 𝑏﷯, 𝑐﷯﷯ + 𝒃﷯, 𝒃﷯, 𝒄﷯﷯ + 𝒂﷯, 𝒃﷯, 𝒂﷯﷯ + 𝒃﷯, 𝒃﷯, 𝒂﷯﷯ + 𝒂﷯, 𝒄﷯, 𝒂﷯﷯ + 𝑏﷯, 𝑐﷯, 𝑎﷯﷯ = 𝑎﷯, 𝑏﷯, 𝑐﷯﷯ + 0 + 0 + 0 + 0 + 𝑏﷯, 𝑐﷯, 𝑎﷯﷯ = 𝑎﷯, 𝑏﷯, 𝑐﷯﷯ + 𝒃﷯, 𝒄﷯, 𝒂﷯﷯ = 𝑎﷯, 𝑏﷯, 𝑐﷯﷯ + 𝒂﷯, 𝒃﷯, 𝒄﷯﷯ = 2 𝑎﷯, 𝑏﷯, 𝑐﷯﷯ = R.H.S Hence proved