Supplementary Exercise Q9

Transcript

Supplementary Exercise Q9 Show that if the vectors 𝑎﷯, 𝑏﷯ and 𝑐﷯ are co-planar, then 𝑎﷯ + 𝑏﷯, 𝑏﷯ + 𝑐 and 𝑐﷯ + 𝑎﷯ are also co-planar Vectors 𝑎﷯, 𝑏﷯, 𝑐﷯ are coplanar if 𝑎﷯ 𝑏﷯ 𝑐﷯﷯ = 0 Finding 𝑎﷯ + 𝑏﷯﷮ 𝑏﷯+ 𝑐﷯﷮ 𝑐﷯+ 𝑎﷯﷯﷯ 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﷯﷯ = 2 × 0 = 0 Since 𝑎﷯ + 𝑏﷯﷮ 𝑏﷯+ 𝑐﷯﷮ 𝑐﷯+ 𝑎﷯﷯﷯ = 0 𝑎﷯ + 𝑏﷯, 𝑏﷯ + 𝑐 and 𝑐﷯ + 𝑎﷯ are also co-planar Hence proved