Ex 11.2, 1 - Chapter 11 Class 12 Three Dimensional Geometry

Transcript

Ex 11.2, 1 Show that the three lines with direction cosines 12﷮13﷯, − 3﷮13﷯, − 4﷮13﷯ ; 4﷮13﷯, 12﷮13﷯, 3﷮13﷯ ; 3﷮13﷯, − 4﷮13﷯, 12﷮13﷯ are mutually perpendicular. Two lines with direction cosines 𝑙﷮1﷯, 𝑚﷮1﷯ , 𝑛﷮1﷯ & 𝑙﷮2﷯, 𝑚﷮2﷯ , 𝑛﷮2﷯ are perpendicular to each other if 𝒍𝟏 𝒍𝟐 + 𝒎𝟏 𝒎𝟐 + 𝒏𝟏 𝒏𝟐 = 0 𝑙1 𝑙2 + 𝑚1 𝑚2 + 𝑛1 𝑛2 = 12﷮13﷯ × 4﷮13﷯﷯ + −3﷮13﷯ × 12﷮13﷯﷯ + −4﷮13﷯ × 3﷮13﷯﷯ = 48﷮169﷯ + −36﷮169﷯﷯ + −12﷮169﷯﷯ = 48 − 36 −12﷮169﷯ = 48 − 48﷮169﷯ = 0 ∴ 𝑙1 𝑙2 + 𝑚1 𝑚2 + 𝑛1 𝑛2 = 0 Hence, the two lines are perpendicular. Now, 𝑙2 𝑙3 + 𝑚2 𝑚3 + 𝑛2 𝑛3 = 4﷮13﷯× 3﷮13﷯﷯ + 12﷮13﷯ × − 4﷮13﷯﷯ + 3﷮13﷯ × 12﷮13﷯﷯ = 12﷮169﷯ + − 48﷮169﷯﷯ + 36﷮169﷯ = 12 − 48 + 36﷮169﷯ = 48 − 48﷮169﷯ = 0 ∴ 𝑙2 𝑙3 + 𝑚2 𝑚3 + 𝑛2 𝑛3 = 0 Hence, the two lines are perpendicular.