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Angle Between Lines and Planes

Angle Between Lines and Planes. Definition. H. E. D. A. G. F. B. C. Identifying Planes. A plane is a flat surface. Examples:. ABCD BCGF CGHD BFEA EFGH. Identify. E. H. The line AC lies on the plane ABCD. F. G. A. D. B. C. Lines on a Plane.

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Angle Between Lines and Planes

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  1. Angle Between Lines and Planes

  2. Definition H E D A G F B C Identifying Planes A plane is a flat surface. Examples: • ABCD • BCGF • CGHD • BFEA • EFGH

  3. Identify E H The line AC lies on the plane ABCD. F G A D B C Lines on a Plane

  4. The line AH lies on the plane ADHE. Lines on a Plane Identify E H F G A D B C

  5. The line AG intersects with the plane EFCD. Lines on a Plane Identify E H F G A D B C

  6. Definition Normal Plane Normalsto a Plane A line which is perpendicular to any line on the plane that passes through the point of intersection of the line with the plane

  7. Definition AOB is the angle between the line OA and the plane PQRS. S R Q P Angle Between Lines and Planes It is the angle between the line and its orthogonal projection on the plane. A O B

  8. Example 1 E H F G A D B C Angle Between Lines and Planes Name the angle between the line BH and the plane BCGF.

  9. Angle Between Lines and Planes Name the angle between the line BH and the plane BCGF. Example 1 Solution: • The line HG is the normal to the plane BCGF. E H • BG is the orthogonal projection of the line BH on the plane BCGF. F G A D •  HBG is the angle between the BH and the plane BCGF. B C

  10. Example 2 Angle Between Lines and Planes Name the angle between the line BH and the plane EFGH. E H F G A D B C

  11. Angle Between Lines and Planes Name the angle between the line BH and the plane EFGH. Example 2 Solution: • The line BF is the normal to the plane EFGH. E H • FH is the orthogonal projection of the line BH on the plane EFGH. F G A D •  BHF is the angle between the BH and the plane EFGH. B C

  12. Example 3 Angle Between Lines and Planes Name the angle between the line BH and the plane ABFE. E H F G A D B C

  13. Angle Between Lines and Planes Name the angle between the line BH and the plane ABFE. Example 3 Solution: • The line EH is the normal to the plane ABFE. E H • BE is the orthogonal projection of the line BH on the plane ABFE. F G A D •  HBE is the angle between the BH and the plane ABFE. B C

  14. Example 4 H E 8 cm F G A D 10 cm B C 12 cm Angle Between Lines and Planes The diagram below shows a model of a cuboid which is made of iron rods. Calculate • the length CE, • the angle between the line CE and the plane BCGF.

  15. Example 4 F E 10 cm 8 cm C F H E B C 12 cm F G Pythagoras’ theorem Pythagoras’ theorem A D B C Angle Between Lines and Planes Solution: (a) In ∆BCF, CF2 = BC2 + BF2 8 cm = 122 + 102 10 cm In ∆FCE, CE2 = CF2 + FE2 12 cm = 122 + 102 + 82 = 308 CE = 17.55 cm

  16. Example 4 E 8 cm C F H E 8 cm F G In ∆FCE, sin ECF = A D 10 cm B C = 12 cm Angle Between Lines and Planes Solution: (b) The angle between the line CE and the plane BCGF is ECF. ECF = 27° 7'

  17. The End

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