Line Plane In 3 Dimension

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Line Plane In 3 Dimension

  1. 1. LINES AND PLANES IN 3-DIMENSION To answer the question from this topic, the students must acquire the following skills : • Able to identify the angle between a line and a plane ( 1 Mark ) • Able to calculate the angle between a line and a plane ( 3 Marks ). • Able to identify the angle between two planes ( 1 Mark ). • Able to calculate the angle between two planes ( 3 Marks )
  2. 2. EXAMPLE : Diagram shows a cuboid with W T P1 a horizontal rectangular base. 5cm  Calculate the angle between U V the planeTWR and the plane S PSWT. P 8cm SOLUTION : Q R  R T/W S 8 Tan  RWS =  5 K2 At the back W 8  RWS = Tan -1 5 5cm S R = 580  N1 8cm
  3. 3. ACTIVITY 1 : TO IDENTIFY THE PLANE H G E F D C A B PLANE AT THE TOP : PLANE EFGH
  4. 4. H G E F D C A B PLANE ON THE LEFT : PLANE ADHE
  5. 5. H G E F D C A B PLANE IN THE FRONT : PLANE ABFE
  6. 6. H G E F D C A B PLANE AT THE BACK : PLANE DCGH
  7. 7. H G E F D C A B PLANE AT THE BOTTOM: PLANE ABCD
  8. 8. H G E F D C A B PLANE ON THE RIGHT : PLANE BCGF
  9. 9. THE LOCATION OF THE POINT ON TOP OF THE RED DOT TO THE RIGHT OF THE RED DOT AT THE BACK OF THE RED DOT IN FRONT OF THE RED DOT
  10. 10. ON THE TOP OF …. AT THE TO THE BACK OF …. LEFT OF …. TO THE RIGHT OF …. IN FRONT OF …. AT THE BOTTOM OF ….
  11. 11. ACTIVITY 2 : TO DETERMINE THE LOCATION OF A POINT H G E F D C A B POINT TO THE LEFT OF F : POINT E POINT AT THE BOTTOM OF F : POINT B POINT AT THE BACK OF F : POINT G POINT TO THE RIGHT OF D : POINT C POINT ON TOP OF D : POINT H POINT IN FRONT OF D : POINT A
  12. 12. ANGLE BETWEEN A LINE AND A PLANE A LINE PLANE C B
  13. 13. Activity 3 :To Identify The Angle Between Line And Plane H G The line draw from G and Normal perpendicular to the plane ABCD is E F call normal D C The line lies on the Orthogonal plane ABCD which projection joint the point A to the A B line GC is known as the orthogonal The angle between the line AG projection of line AG and the orthogonal projection, AC on the plane ABCD. is the angle between the line AG and the plane ABCD that is  GAC.
  14. 14. ACTIVITY 3 : To Identify The Angle Between A Line And A Plane Example 1a  H G G A C Normal E F D C At the bottom Name the angle between the line A B AG and the plane ABCD Orthogonal projection Angle between the line AG and the plane ABCD = GAC.
  15. 15. EXAMPLE 1(b) H G E F D C A B Diagram 1(b) Diagram 1b shows a cuboid ABCDEFGH. Name the angle between the line HB and the plane ABCD.
  16. 16. ACTIVITY 4 : To find the angle between a line and a plane Example 2(a) 12cm H G 5cm 5cm E F D C 4cm A B Diagram 2a Diagram 2(a) shows a cuboid, ABCDEFG. Find the angle between the line AH and the plane DCGH.
  17. 17. No Steps Solutions 1. Draw the line AH and shade the plan DCGH H 12cm G in diagram 2a. 5cm E F D C 4cm A B Diagram 2a Diagram 2a shows a cuboid, ABCDEFG. Find the angle between the line AH and the plane DCGH.
  18. 18. No Steeps Solutions 2 Use the method you have learned in activity 3, identify the angle between the line AH  A H D and the plane DCGH back H 12cm G 5cm E F D C 4cm A B
  19. 19. No Steps Solutions 3 Refer to the points you have H obtained in steep 2 (point A, H, D), complete the ∆ AHD. Mark  AHD. Mark the right angle,  HDA. Transfer out the A D ∆ AHD. 12cm  H G A H D 5cm E F D C 4cm A B
  20. 20. No Steps Solutions 4 With the information given in the question, label the length of the sides of ∆ AHD. At least the length for 2 sides must be known. Use Pythegoras Theorem if necessary. H 12cm G 5cm E F D C 4cm A B
  21. 21. No Steps Solutions 6 Mark, H - the opposite side, AD asT - the adjacent side, HD as S 5 cm S A 4 cm D T H 12cm G 5cm E F D C 4cm A B
  22. 22. No Steps Solutions 6 Use the tangent formula to 4 calculate  AHD. Tan  AHD = 5 Remember, use 4 -The sine formula, if O and H were  AHD = tan -1 5 known O - SOH S  AHD = 38040’ H - The cosine formula, if A and H were known A C – CAH 12cm H G H -The tangent formula, if O and A E 5cm D F were known O C T – TOA 4cm A A B
  23. 23. example 2 (b) 12 cm H G E F 4 cm D C 3 cm A B Diagram 2b Diagram 2b shows a cuboid,ABCDEFGH. Calculate the angle between the line HB and the plane BCGF
  24. 24. ANGLE BETWEEN TWO PLANES
  25. 25. ACTIVITY 5 : To Identified The Angle Between Two Planes EXAMPLE 3(a) H G E F D C A 1. DRAW 3 LINES B Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane AGH and the plane ABCD
  26. 26. ACTIVITY 5 : To Identified The Angle Between Two Planes H G E F D C Bottom A B 2. Mark the location Diagram 3a (direction) of the plane ABCD at the Diagram 3a shows a cuboid, ABCDEFGH. Name the angle bottom of the first between the plane, AGH and the line to the left. plane, ABCD
  27. 27. ACTIVITY 5 : To Identified The Angle Between Two Planes H G A E F D C Bottom A B 3. Refer to the plane, Diagram 3a AGH, identify the points which Diagram 3a shows a cuboid, touch the plane, ABCDEFGH. Name the angle between the plane, AGH and the ABCD and write it plane, ABCD at the middle line.
  28. 28. ACTIVITY 5 : To Identified The Angle Between Two Planes H G H/G A E F D C Bottom A 3. Refer to the plane, B Diagram 3a AGH, identify the point which does Diagram 3a shows a cuboid, not touch the ABCDEFGH. Name the angle plane, ABCD and between the plane, AGH and the write it at the first plane, ABCD line to the left.
  29. 29. ACTIVITY 5 : To Identified The Angle Between Two Planes H G H/G A E Bottom F D C 5. Between the point H and G, point which is A B nearer to point A or Diagram 3a located on the same plane as point A will Diagram 3a shows a cuboid, ABCDEFGH. Name the angle be choosen. Point between the plane, AGH and the which is not choosen plane, ABCD will be earased.
  30. 30. ACTIVITY 5 : To Identified The Angle Between Two Planes H G H A E Ke Bawah F D C 5. Between the point H and G, point which is A B nearer to point A or Diagram 3a located on the same plane as point A will Diagram 3a shows a cuboid, ABCDEFGH. Name the angle be choosen. Point between the plane, AGH and the which is not choosen plane, ABCD will be earased.
  31. 31. ACTIVITY 5 : To Identified The Angle Between Two Planes H G  H A D E Bottom F D C 6. Identify the point which is located at A B the bottom of the Diagram 3a point H ( )and write it on the first Diagram 3a shows a cuboid, line to the right. ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD
  32. 32. ACTIVITY 5 : To Identified The Angle Between Two Planes G H  H A D E F D C Bottom A B 7. In the diagram 3a, Diagram 3a complete the ∆ HAD and mark the  HAD  Angle between the plane, AGH and the plane, ABCD =  HAD
  33. 33. EXAMPLE 3(b) H 12cm G 5cm E F D C 4cm A B Diagram 3b Diagram 3b shows a cuboid with horizontal rectangle base ABCD. Name the angle between the plane ACH and the plane CDHG
  34. 34. THANK YOU

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