Ex 11.2, 8

Transcript

Ex 11.2, 8 Find the vector and the Cartesian equations of the lines that passes through the origin and (5, −2,3). Vector equation Vector equation of a line passing though two points with position vectors 𝑎﷯ and 𝑏﷯ is 𝑟﷯ = 𝑎 ﷯ + 𝜆 ( 𝑏﷯ − 𝑎﷯) Given, Let two points be A (0, 0, 0) & B(5, − 2,3) So, 𝑟﷯ = (0 𝑖﷯ + 0 𝑗﷯ + 0 𝑘﷯) + 𝜆 5 𝑖﷯−2 𝑗﷯+3 𝑘﷯﷯ − 0 𝑖﷯+0 𝑗﷯+0 𝑘﷯﷯﷯ 𝒓﷯ = 𝜆 (5 𝒊﷯ − 2𝒋 + 3 𝒌﷯) Cartesian equation Cartesian equation of a line passing through two points A (x1, y1, z1) and B (x2, y2, z2) is 𝑥 − 𝑥﷮1﷯﷮ 𝑥﷮2﷯ − 𝑥﷮1﷯﷯ = 𝑦 − 𝑦﷮1﷯﷮ 𝑦﷮2﷯ − 𝑦﷮1﷯﷯ = 𝑧 − 𝑧﷮1﷯﷮ 𝑧﷮2﷯ − 𝑧﷮1﷯﷯ Since the line passes through A (0, 0, 0) x1 = 0, y1 = 0, z1 = 0 And also passes through B (5, −2, 3), x2 = 5, y2 = − 2, z2 = 3 Equation of line is 𝑥 − 0﷮5 − 0﷯ = 𝑦 − 0﷮ −2 − 0﷯ = 𝑧 − 0﷮3 − 0﷯ 𝒙﷮𝟓﷯ = 𝒚﷮ −𝟐﷯ = 𝒛﷮𝟑﷯