Supplementary Exercise Q4

Transcript

Supplementary Exercise Q4 Show that (i) the vectors 𝑎﷯ = 2 𝑖﷯ − 𝑗﷯ + 𝑘﷯, 𝑏﷯ = 𝑖﷯ + 2 𝑗﷯ − 3 𝑘﷯, and 𝑐﷯ = 3 𝑖﷯ − 4 𝑗﷯ + 5 𝑘﷯ are coplanar. Three vectors 𝑎﷯, 𝑏﷯, 𝑐﷯ are coplanar if [ 𝒂﷯, 𝒃﷯, 𝒄﷯ ] = 0 Given, 𝑎﷯ = 2 𝑖﷯ − 𝑗﷯ + 𝑘﷯ 𝑏﷯ = 𝑖﷯ + 2 𝑗﷯ − 3 𝑘﷯ 𝑐﷯ = 3 𝑖﷯ − 4 𝑗﷯ + 5 𝑘﷯ 𝑎﷯,𝑏, 𝑐﷯﷯ = 2﷮−1﷮1﷮1﷮2﷮−3﷮3﷮−4﷮5﷯﷯ = 2 2×5﷯−(−4×−3)﷯ − (−1) 1×5﷯−(3×−3) ﷯ + 1 1×−4﷯−(3×2) ﷯ = 2 10−12﷯+ 5+9﷯ + −4−6﷯ = –4 + 14 – 10 = 0 ∴ 𝒂﷯,𝒃, 𝒄﷯﷯ = 0 Therefore, 𝑎﷯,𝑏 and 𝑐﷯ are coplanar. Supplementary Exercise Q4 Show that (ii) the vectors 𝑎﷯ = 𝑖﷯ − 2 𝑗﷯ + 3 𝑘﷯, 𝑏﷯ = −2 𝑖﷯ + 3 𝑗﷯ − 4 𝑘﷯, and 𝑐﷯ = 𝑖﷯ − 3 𝑗﷯ + 5 𝑘﷯ are coplanar Three vectors 𝑎﷯, 𝑏﷯, 𝑐﷯ are coplanar if [ 𝒂﷯, 𝒃﷯, 𝒄﷯ ] = 0 Given, 𝑎﷯ = 𝑖﷯ − 2 𝑗﷯ + 3 𝑘﷯ 𝑏﷯ = −2 𝑖﷯ + 3 𝑗﷯ − 4 𝑘﷯ 𝑐﷯ = 𝑖﷯ − 3 𝑗﷯ + 5 𝑘﷯ 𝑎﷯,𝑏, 𝑐﷯﷯ = 1﷮−2﷮3﷮−2﷮3﷮−4﷮1﷮−3﷮5﷯﷯ = 1 3×5﷯−(−3×4) ﷯ − (−2) −2×5﷯−(1×−4) ﷯ + 3 −2×−3﷯−(1×3) ﷯ = 1 15−12﷯+ 2 −10−(−4﷯﷯ + 3 6−3﷯ = 1(3) + 2 (−6) + 3 (3) = 3 − 12 + 9 = 12 − 12 = 0 ∴ 𝒂﷯,𝒃, 𝒄﷯﷯ = 0 Therefore, 𝑎﷯,𝑏 and 𝑐﷯ are coplanar.