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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 = 43 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) = ( 𝑥1× 𝑎) + (𝑦1 × 𝑎) + (𝑧1 × 𝑎) = 𝑎x1 + ay1 + az1 Magnitude of 𝑂𝑃 = 𝑥12+ 𝑦12+ 𝑧12 𝑂𝑃 = 𝑥12+ 𝑦12+ 𝑧12 Magnitude of 𝑂𝐸 = 𝑎2+𝑎2+𝑎2 𝑂𝐸 = 3𝑎2 = 3𝑎 So, cos α = 𝑎𝑥1 + 𝑎𝑦1 + 𝑎𝑧1 𝑥12 + 𝑦12 + 𝑧12 × 3𝑎 = 𝑎(𝑥1 + 𝑦1 + 𝑧1) 𝑥12 + 𝑦12 + 𝑧12 × 3𝑎 = (𝒙𝟏+𝒚𝟏 + 𝒛𝟏) 𝟑 𝒙𝟏𝟐 + 𝒚𝟏𝟐 + 𝒛𝟏𝟐 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 = 𝑥12+ 𝑦12+𝑧12 Magnitude of 𝐴𝐹 = −𝑎2+𝑎2+𝑎2 𝐴𝐹 = 𝑎2+𝑎2+𝑎2 = 3𝑎2 = 3𝑎 So, cos 𝜷 = 𝑎(−𝑥1+ 𝑦1 + 𝑧1) 𝑥12 + 𝑦12 + 𝑧12 × 3𝑎 = −𝒙𝟏+ 𝒚𝟏 + 𝒛𝟏 𝟑 𝒙𝟏𝟐 + 𝒚𝟏𝟐 + 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 = 𝑥12+ 𝑦12+𝑧12 Magnitude of 𝐶𝐷 = 𝑎2+𝑎2+ −𝑎2 𝐶𝐷 = 𝑎2+𝑎2+𝑎2 = 3𝑎2 = 3𝑎 So, cos 𝜸 = 𝑎(𝑥1+ 𝑦1 − 𝑧1) 𝑥12 + 𝑦12 + 𝑧12 × 3𝑎 = 𝒙𝟏+ 𝒚𝟏 − 𝒛𝟏 𝟑 𝒙𝟏𝟐 + 𝒚𝟏𝟐 + 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 = 𝑥12+ 𝑦12+𝑧12 Magnitude of 𝐵𝐺 = 𝑎2+ −𝑎2+𝑎2 𝐵𝐺 = 𝑎2+𝑎2+𝑎2 = 3𝑎2 = 3𝑎 So, cos 𝜹 = 𝑎(𝑥1− 𝑦1 + 𝑧1) 𝑥12 + 𝑦12 + 𝑧12 × 3𝑎 = 𝒙𝟏 − 𝒚𝟏 + 𝒛𝟏 𝟑 𝒙𝟏𝟐 + 𝒚𝟏𝟐 + 𝒛𝟏𝟐 Now, cos2 𝜶 + cos2𝜷 + cos2 𝜸 + cos2 𝜹 = 𝑥1 + 𝑦1 + 𝑧1 3 𝑥12 + 𝑦12 + 𝑧122 + − 𝑥1 + 𝑦1 + 𝑧1 3 𝑥12 + 𝑦12 + 𝑧122 + 𝑥1 + 𝑦1 − 𝑧1 3 𝑥12 + 𝑦12 + 𝑧122 + 𝑥1 − 𝑦1 + 𝑧1 3 𝑥12 + 𝑦12 + 𝑧122 = 𝑥1 + 𝑦1 + 𝑧123( 𝑥12 + 𝑦12 + 𝑧12) + − 𝑥1 + 𝑦1 + 𝑧123( 𝑥12 + 𝑦12 + 𝑧12) + 𝑥1 + 𝑦1 − 𝑧123( 𝑥12 + 𝑦12 + 𝑧12) + 𝑥1 − 𝑦1 + 𝑧123( 𝑥12 + 𝑦12 + 𝑧12) = 𝑥1 + 𝑦1 + 𝑧12 + −𝑥1 + 𝑦1+ 𝑧12 + 𝑥1 + 𝑦1 − 𝑧12 + 𝑥1 − 𝑦1 + 𝑧123( 𝑥12 + 𝑦12 + 𝑧12) = 𝑥12 + 𝑦12 + 𝑧12 + 2 𝑥1 𝑦1+ 2 𝑥1 𝑧1+ 2 𝑦1 𝑧1 + 𝑥12 + 𝑦12 + 𝑧12 − 2 𝑥1 𝑦1 − 2 𝑥1 𝑧1+ 2 𝑦1 𝑧1+ 𝑥12 + 𝑦12 + 𝑧12+ 2 𝑥1 𝑦1 − 2 𝑥1 𝑧1 − 2 𝑦1 𝑧1+ 𝑥12 + 𝑦12 + 𝑧12 − 2 𝑥1 𝑦1 + 2 𝑥1 𝑧1 − 2 𝑦1 𝑧13( 𝑥12 + 𝑦12 + 𝑧12) = 4 𝑥12 + 4 𝑦12 + 4 𝑧123( 𝑥12 + 𝑦12 + 𝑧12) = 4( 𝑥12 + 𝑦12 + 𝑧12)3( 𝑥12 + 𝑦12 + 𝑧12) = 43 ∴ cos2𝜶 + cos2𝜷 + cos2𝜸 + cos2𝜹 = 𝟒𝟑

Examples

Example 1

Example, 2 Important

Example, 3

Example, 4 Important

Example, 5 Important

Example, 6 Important

Example, 7

Example 8

Example, 9 Deleted for CBSE Board 2022 Exams

Example 10 Important Deleted for CBSE Board 2022 Exams

Example 11

Example 12 Important

Example 13 Important

Example 14

Example 15

Example 16 Important

Example 17

Example 18

Example 19 Important

Example 20 Important

Example 21 Important

Example 22 Deleted for CBSE Board 2022 Exams

Example 23 Important Deleted for CBSE Board 2022 Exams

Example 24

Example, 25 Important Deleted for CBSE Board 2022 Exams

Example 26 You are here

Example 27 Important

Example 28 Important

Example 29 Important

Example 30 Important

Chapter 11 Class 12 Three Dimensional Geometry (Term 2)

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.