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Example 26 - Chapter 11 3D Geometry Class 12 - A line makes - Direction cosines and ratios

Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3 Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 4 Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 5 Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 6 Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 7 Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 8 Example 26 - Chapter 11 Class 12 Three Dimensional Geometry - Part 9

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Example 26 A line makes angles 𝛼, 𝛽 , 𝛾 and 𝛿 with the diagonals of a cube, prove that cos2 𝛼 + cos2𝛽 + cos2 𝛼 + cos2𝛾 = 4﷮3﷯ Let the side of a cube be ‘a’ Vertices of the cube are ABCDEFOG as shown. Consider a line OP , P having the coordinate (x1, y1, z1) Let OP make angles 𝛼, 𝛽 , 𝛾 and 𝛿 with diagonals OE, AF, CD and BG respectively. Finding lines OE, AF, CD and BG first. Now, Angle between 𝑂𝑃﷯ and 𝑂𝐸﷯ is given by cos 𝛼 = 𝑂𝑃﷯. 𝑂𝐸﷯﷮ 𝑂𝑃﷯﷯ 𝑂𝐸﷯﷯﷯ 𝑂𝑃﷯. 𝑂𝐸﷯ = (x1 𝑖﷯ + y1 𝑗﷯ + z1 𝑘﷯) . (𝑎 𝑖﷯ + 𝑎 𝑗﷯ + 𝑎 𝑘﷯) = ( 𝑥﷮1﷯× 𝑎) + (𝑦1 × 𝑎) + (𝑧1 × 𝑎) = 𝑎x1 + ay1 + az1 Magnitude of 𝑂𝑃﷯ = ﷮ 𝑥﷮1﷯﷮2﷯+ 𝑦﷮1﷯﷮2﷯+ 𝑧﷮1﷯﷮2﷯﷯ 𝑂𝑃﷯﷯ = ﷮ 𝑥﷮1﷯﷮2﷯+ 𝑦﷮1﷯﷮2﷯+ 𝑧﷮1﷯﷮2﷯﷯ Magnitude of 𝑂𝐸﷯ = ﷮𝑎2+𝑎2+𝑎2﷯ 𝑂𝐸﷯﷯ = ﷮3𝑎2﷯ = ﷮3﷯𝑎 So, cos α = 𝑎𝑥1 + 𝑎𝑦1 + 𝑎𝑧1﷮ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧﷮1﷮2﷯﷯ × ﷮3﷯𝑎﷯ = 𝑎(𝑥1 + 𝑦1 + 𝑧1)﷮ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧﷮1﷮2﷯﷯ × ﷮3﷯𝑎﷯ = (𝒙𝟏+𝒚𝟏 + 𝒛𝟏)﷮ ﷮𝟑﷯ ﷮ 𝒙﷮𝟏﷮𝟐﷯ + 𝒚﷮𝟏﷮𝟐﷯ + 𝒛﷮𝟏﷮𝟐﷯﷯ ﷯ Angle between 𝑂𝑃﷯ and 𝐴𝐹﷯ is given by cos 𝛽 = 𝑂𝑃﷯. 𝐴𝐹﷯﷮ 𝑂𝑃﷯﷯ 𝐴𝐹﷯﷯﷯ 𝑂𝑃﷯. 𝐴𝐹﷯ = (x1 𝑖﷯ + y1 𝑗﷯ + z1 𝑘﷯).(−𝑎 𝑖﷯ + 𝑎 𝑗﷯ + 𝑎 𝑘﷯) = (x1× –𝑎) + (y1 × 𝑎) + (z1 × 𝑎) = –𝑎x1 + ay1 + az1 Magnitude of 𝑂𝑃﷯ = 𝑂𝑃﷯﷯ = ﷮ 𝑥﷮1﷮2﷯+ 𝑦﷮1﷮2﷯+𝑧12﷯ Magnitude of 𝐴𝐹﷯ = ﷮ −𝑎﷯2+𝑎2+𝑎2﷯ 𝐴𝐹﷯﷯ = ﷮𝑎2+𝑎2+𝑎2 ﷯ = ﷮3𝑎2﷯ = ﷮3﷯𝑎 So, cos 𝜷 = 𝑎(−𝑥1+ 𝑦1 + 𝑧1)﷮ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12﷯ × ﷮3﷯𝑎﷯ = −𝒙𝟏+ 𝒚𝟏 + 𝒛𝟏﷮ ﷮𝟑﷯ ﷮ 𝒙﷮𝟏﷮𝟐﷯ + 𝒚﷮𝟏﷮𝟐﷯ + 𝒛𝟏𝟐﷯﷯ Angle between 𝑂𝑃﷯ and 𝐶𝐷﷯ is given by cos 𝛾 = 𝑂𝑃﷯. 𝐶𝐷﷯﷮ 𝑂𝑃﷯﷯ 𝐶𝐷﷯﷯﷯ 𝑂𝑃﷯. 𝐶𝐷﷯ = (x1 𝑖﷯ + y1 𝑗﷯ + z1 𝑘﷯).(𝑎 𝑖﷯ + 𝑎 𝑗﷯ – 𝑎 𝑘﷯) = (x1× 𝑎) + (y1 × 𝑎) + (z1 × –𝑎) = 𝑎x1 + ay1 – az1 Magnitude of 𝑂𝑃﷯ = 𝑂𝑃﷯﷯ = ﷮ 𝑥﷮1﷮2﷯+ 𝑦﷮1﷮2﷯+𝑧12﷯ Magnitude of 𝐶𝐷﷯ = ﷮𝑎2+𝑎2+ −𝑎﷯2﷯ 𝐶𝐷﷯﷯ = ﷮𝑎2+𝑎2+𝑎2 ﷯ = ﷮3𝑎2﷯ = ﷮3﷯𝑎 So, cos 𝜸 = 𝑎(𝑥1+ 𝑦1 − 𝑧1)﷮ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12﷯ × ﷮3﷯𝑎﷯ = 𝒙𝟏+ 𝒚𝟏 − 𝒛𝟏﷮ ﷮𝟑﷯ ﷮ 𝒙﷮𝟏﷮𝟐﷯ + 𝒚﷮𝟏﷮𝟐﷯ + 𝒛𝟏𝟐﷯﷯ Angle between 𝑂𝑃﷯ and 𝐵𝐺﷯ is given by cos 𝛿 = 𝑂𝑃﷯. 𝐵𝐺﷯﷮ 𝑂𝑃﷯﷯ 𝐵𝐺﷯﷯﷯ 𝑂𝑃﷯. 𝐵𝐺﷯ = (x1 𝑖﷯ + y1 𝑗﷯ + z1 𝑘﷯).(𝑎 𝑖﷯ – 𝑎 𝑗﷯ + 𝑎 𝑘﷯) = (x1× 𝑎) + (y1 × –𝑎) + (z1 × 𝑎) = 𝑎x1 – ay1 + az1 Magnitude of 𝑂𝑃﷯ = 𝑂𝑃﷯﷯ = ﷮ 𝑥﷮1﷮2﷯+ 𝑦﷮1﷮2﷯+𝑧12﷯ Magnitude of 𝐵𝐺﷯ = ﷮𝑎2+ −𝑎﷯2+𝑎2﷯ 𝐵𝐺﷯﷯ = ﷮𝑎2+𝑎2+𝑎2 ﷯ = ﷮3𝑎2﷯ = ﷮3﷯𝑎 So, cos 𝜹 = 𝑎(𝑥1− 𝑦1 + 𝑧1)﷮ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12﷯ × ﷮3﷯𝑎﷯ = 𝒙𝟏 − 𝒚𝟏 + 𝒛𝟏﷮ ﷮𝟑﷯ ﷮ 𝒙﷮𝟏﷮𝟐﷯ + 𝒚﷮𝟏﷮𝟐﷯ + 𝒛𝟏𝟐﷯﷯ Now, cos2 𝜶 + cos2𝜷 + cos2 𝜸 + cos2 𝜹 = 𝑥1 + 𝑦1 + 𝑧1﷮ ﷮3﷯ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12﷯﷯﷯﷮2﷯ + − 𝑥1 + 𝑦1 + 𝑧1﷮ ﷮3﷯ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12﷯﷯﷯﷮2﷯ + 𝑥1 + 𝑦1 − 𝑧1﷮ ﷮3﷯ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12﷯﷯﷯﷮2﷯ + 𝑥1 − 𝑦1 + 𝑧1﷮ ﷮3﷯ ﷮ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12﷯﷯﷯﷮2﷯ = 𝑥1 + 𝑦1 + 𝑧1﷯﷮2﷯﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ + − 𝑥1 + 𝑦1 + 𝑧1﷯2﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ + 𝑥1 + 𝑦1 − 𝑧1﷯2﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ + 𝑥1 − 𝑦1 + 𝑧1﷯2﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ = 𝑥1 + 𝑦1 + 𝑧1﷯2 + −𝑥1 + 𝑦1+ 𝑧1﷯2 + 𝑥1 + 𝑦1 − 𝑧1﷯2 + 𝑥1 − 𝑦1 + 𝑧1﷯2﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ = 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧﷮1﷮2﷯ + 2 𝑥﷮1﷯ 𝑦﷮1﷯+ 2 𝑥﷮1﷯ 𝑧﷮1﷯+ 2 𝑦﷮1﷯ 𝑧﷮1﷯ + 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧﷮1﷮2﷯ − 2 𝑥﷮1﷯ 𝑦﷮1﷯ − 2 𝑥﷮1﷯ 𝑧﷮1﷯+ 2 𝑦﷮1﷯ 𝑧﷮1﷯﷮+ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧﷮1﷮2﷯+ 2 𝑥﷮1﷯ 𝑦﷮1﷯ − 2 𝑥﷮1﷯ 𝑧﷮1﷯ − 2 𝑦﷮1﷯ 𝑧﷮1﷯+ 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧﷮1﷮2﷯ − 2 𝑥﷮1﷯ 𝑦﷮1﷯ + 2 𝑥﷮1﷯ 𝑧﷮1﷯ − 2 𝑦﷮1﷯ 𝑧﷮1﷯﷯﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ = 4 𝑥﷮1﷮2﷯ + 4 𝑦﷮1﷮2﷯ + 4 𝑧﷮1﷮2﷯﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ = 4( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷮3( 𝑥﷮1﷮2﷯ + 𝑦﷮1﷮2﷯ + 𝑧12)﷯ = 4﷮3﷯ ∴ cos2𝜶 + cos2𝜷 + cos2𝜸 + cos2𝜹 = 𝟒﷮𝟑﷯

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.