Ex 11.2, 15

Transcript

Ex 11.2, 15 (Cartesian method) Find the shortest distance between the lines 𝑥 + 1﷮7﷯ = 𝑦 + 1﷮ − 6﷯ = 𝑧 + 1﷮1﷯ and 𝑥 − 3﷮1﷯ = 𝑦 − 5﷮ − 2﷯ = 𝑧 − 7﷮1﷯ Shortest distance between two lines l1: 𝑥 − 𝑥﷮1﷯﷮ 𝑎﷮1﷯﷯ = 𝑦 − 𝑦﷮1﷯﷮ 𝑏﷮1﷯﷯ = 𝑧 − 𝑧﷮1﷯﷮ 𝑐﷮1﷯﷯ l2: 𝑥 − 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ = 𝑦 − 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯ = 𝑧 − 𝑧﷮2﷯﷮ 𝑐﷮2﷯﷯ is 𝒙﷮𝟐﷯ − 𝒙﷮𝟏﷯﷮ 𝒚﷮𝟐﷯ − 𝒚﷮𝟏﷯﷮ 𝒛﷮𝟐﷯ − 𝒛﷮𝟏﷯﷮ 𝒂﷮𝟏﷯﷮ 𝒃﷮𝟏﷯﷮ 𝒄﷮𝟏﷯﷮ 𝒂﷮𝟐﷯﷮ 𝒃﷮𝟐﷯﷮ 𝒄﷮𝟐﷯﷯﷯﷮ ﷮ 𝒂﷮𝟏﷯ 𝒃﷮𝟐﷯ − 𝒂﷮𝟐﷯ 𝒃﷮𝟏﷯﷯﷮𝟐﷯ + 𝒃﷮𝟏﷯ 𝒄﷮𝟐 ﷯− 𝒃﷮𝟐﷯ 𝒄﷮𝟏﷯﷯﷮𝟐﷯ + 𝒄﷮𝟏﷯ 𝒂﷮𝟐﷯ − 𝒄﷮𝟐﷯ 𝒂﷮𝟏﷯﷯﷮𝟐﷯ ﷯﷯﷯ d = 𝑥﷮2﷯− 𝑥﷮1﷯﷮ 𝑦﷮2﷯ − 𝑦﷮1﷯﷮ 𝑧﷮2﷯ − 𝑧﷮1﷯﷮ 𝑎﷮1﷯﷮ 𝑏﷮1﷯﷮ 𝑐﷮1﷯﷮ 𝑎﷮2﷯﷮ 𝑏﷮2﷯﷮ 𝑐﷮2﷯﷯﷯﷮ ﷮ 𝑎﷮1﷯ 𝑏﷮2﷯ − 𝑎﷮2﷯ 𝑏﷮1﷯﷯﷮2﷯ + 𝑏﷮1﷯ 𝑐﷮2 ﷯− 𝑏﷮2﷯ 𝑐﷮1﷯﷯﷮2﷯ + 𝑐﷮1﷯ 𝑎﷮2﷯ − 𝑐﷮2﷯ 𝑎﷮1﷯﷯﷮2﷯ ﷯﷯﷯ d = 3−(−1)﷮5−(−1)﷮7−(−1)﷮7﷮−6﷮1﷮1﷮−2﷮1﷯﷯﷮ ﷮ 7(−2) −1(−6)﷯﷮2﷯ + −6(1)− −2﷯1﷯﷮2﷯ + 1(1) −1(7)﷯﷮2﷯﷯﷯﷯ d = 4﷮6﷮8﷮7﷮−6﷮1﷮1﷮−2﷮1﷯﷯﷮ ﷮ −14 + 6﷯﷮2﷯ + −6 + 2﷯﷮2﷯ + 1 − 7﷯﷮2﷯﷯﷯﷯ d = 4﷮6﷮8﷮7﷮−6﷮1﷮1﷮−2﷮1﷯﷯﷮ ﷮ 8﷯﷮2﷯ + −4﷯﷮2﷯ + −6﷯﷮2﷯﷯﷯﷯ d = 4﷮6﷮8﷮7﷮−6﷮1﷮1﷮−2﷮1﷯﷯﷮ ﷮116﷯﷯﷯ d = 4 −6 1﷯− −2﷯1﷯−6 7 1﷯−1 1﷯﷯+8 7 −2﷯−1 −6﷯﷯﷮ ﷮116﷯﷯﷯ d = 4 −6 + 2﷯−6 7 − 1﷯+8(−14 + 6)﷮ ﷮116﷯﷯﷯ d = −16 − 36 − 64﷮ ﷮116﷯﷯﷯ d = −116﷮ ﷮116﷯﷯﷯ d = − ﷮116﷯﷯ d = ﷮116﷯ d = ﷮4 × 29﷯ d = 𝟐 ﷮𝟐𝟗﷯ Ex 11.2, 15 (Vector method) Find the shortest distance between the lines 𝑥 + 1﷮7﷯ = 𝑦 + 1﷮ − 6﷯ = 𝑧 + 1﷮1﷯ and 𝑥 − 3﷮1﷯ = 𝑦 − 5﷮ − 2﷯ = 𝑧 − 7﷮1﷯ Shortest distance between two lines 𝑟﷯ = 𝑎1﷯ + λ 𝑏1﷯ and 𝑟﷯ = 𝑎2﷯ + μ 𝑏2﷯ is 𝒃𝟏﷯ × 𝒃𝟐﷯ ﷯. 𝒂𝟐﷯ × 𝒂𝟏﷯ ﷯﷮ 𝒃𝟏﷯ × 𝒃𝟐﷯﷯﷯﷯