Supplementary Example 4

Transcript

Supplementary Example 4 Show that the vectors 𝑎﷯ = 𝑖﷯ − 2 𝑗﷯ + 3 𝑘﷯, 𝑏﷯ = −2 𝑖﷯ + 3 𝑗﷯ − 4 𝑘﷯ and 𝑐﷯ = 𝑖﷯ − 3 𝑗﷯ + 𝜆 𝑘﷯ are co-planar if 𝜆 = 5 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﷯ + λ 𝑘﷯ 𝑎﷯ 𝑏﷯ 𝑐﷯﷯ = 1﷮−2﷮3﷮−2﷮3﷮−4﷮1﷮−3﷮𝜆﷯﷯ 0 = 1 3×λ﷯−(−3×−4) ﷯ − (−2) −2×λ﷯−(1×−4) ﷯ + 3 2×−3﷯−(1×3) ﷯ 0 = 1 3λ−12﷯+2 −2λ−(−4)﷯+3 6−3﷯ 0 = 3λ – 12 – 4λ + 8 + 9 0 = – λ + 5 𝝀 = 5 Therefore, 𝑎﷯,𝑏, 𝑐﷯ are coplanar if λ = 5