Example 29 - Chapter 11 Class 12 - Show lines are coplanar

Example 29 Show that the lines 𝑥 − 𝑎 + 𝑑﷮𝛼 − 𝛿﷯ = 𝑦 − 𝑎﷮𝛼﷯ = 𝑧 − 𝑎 − 𝑑﷮𝛼 + 𝛿﷯ and 𝑥 − 𝑏 + 𝑐﷮𝛽 − 𝛾﷯ = 𝑦 − 𝑏﷮𝛽﷯ = 𝑧 − 𝑏 − 𝑐﷮𝛽 + 𝛾﷯ are coplanar. Two lines 𝑥 − 𝑥﷮1﷯﷮ 𝑎﷮1﷯﷯ = 𝑦 − 𝑦﷮1﷯﷮ 𝑏﷮1﷯﷯ = 𝑧 − 𝑧﷮1﷯﷮ 𝑐﷮1﷯﷯ and 𝑥 − 𝑥﷮2﷯﷮ 𝑎﷮2﷯﷯ = 𝑦 − 𝑦﷮2﷯﷮ 𝑏﷮2﷯﷯ = 𝑧 − 𝑧﷮2﷯﷮ 𝑐﷮2﷯﷯ are coplanar if 𝒙﷮𝟐﷯− 𝒙﷮𝟏﷯﷮ 𝒚﷮𝟐﷯− 𝒚﷮𝟏﷯﷮ 𝒛﷮𝟐﷯− 𝒛﷮𝟏﷯﷮ 𝒂﷮𝟏﷯﷮ 𝒃﷮𝟏﷯﷮ 𝒄﷮𝟏﷯﷮ 𝒂﷮𝟐﷯﷮ 𝒃﷮𝟐﷯﷮ 𝒄﷮𝟐﷯﷯﷯ = 0 Now, 𝑥﷮2﷯− 𝑥﷮1﷯﷮ 𝑦﷮2﷯− 𝑦﷮1﷯﷮ 𝑧﷮2﷯− 𝑧﷮1﷯﷮ 𝑎﷮1﷯﷮ 𝑏﷮1﷯﷮ 𝑐﷮1﷯﷮ 𝑎﷮2﷯﷮ 𝑏﷮2﷯﷮ 𝑐﷮2﷯﷯﷯ = 𝑏−𝑐−𝑎 + 𝑑﷮𝑏−𝑎﷮𝑏+𝑐−𝑎−𝑑﷮𝛼−𝛿﷮𝛼﷮𝛼+𝛿﷮𝛽−𝛾﷮𝛽﷮𝛽+𝛾﷯﷯ Adding column 3 to column 1, = 2(𝑏−𝑎)﷮𝑏−𝑎﷮𝑏+𝑐−𝑎−𝑑﷮2𝛼﷮𝛼﷮𝛼+𝛿﷮2𝛽﷮𝛽﷮𝛽+𝛾﷯﷯ Taking 2 common from Column 1 = 2 𝑏 − 𝑎﷮𝑏 − 𝑎﷮𝑏 + 𝑐 − 𝑎 − 𝑑﷮𝛼﷮𝛼﷮𝛼 + 𝛿﷮𝛽﷮𝛽﷮𝛽 +𝛾﷯﷯ = 2 × 0 = 0 Therefore, the given two lines are coplanar .