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Mathematics

Mathematics. Session. Three Dimensional Geometry–1(Straight Line). Session Objectives. Equation: Passing Through a Fixed Point and Parallel to a Given Vector. Equation: Passing Through Two Fixed Points. Co-linearity of Three Points. Angle Between Two Lines.

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Mathematics

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  1. Mathematics

  2. Session Three Dimensional Geometry–1(Straight Line)

  3. Session Objectives • Equation: Passing Through a Fixed Point and Parallel to a Given Vector • Equation: Passing Through Two Fixed Points • Co-linearity of Three Points • Angle Between Two Lines • Intersection of two lines, Perpendicular distance, Image of a Point • Shortest Distance Between Two Lines • Class Exercise

  4. Z P A Y O X Equation of a Line Passing Through a Fixed Point and Parallel to a Given Vector

  5. Cartesian Form

  6. Example-1

  7. Example –2

  8. Solution Cont.

  9. Z P B A Y O X Passing Through Two Fixed Points

  10. Cartesian Form

  11. Example –3 Find the vector and the Cartesian equations for the line through the points A(3, 4, -7) and B(5, 1, 6).

  12. Solution Cont.

  13. Let P be the point where the line AB crosses the y z-plane. Then, the position vector of the point P is Example –4 Find the coordinates of the points where the line through A(5, 1, 6) and B(3, 4, 1) crosses the y z- plane. Solution: The vector equation of the line through the points A and B is

  14. Solving (ii), (iii) and (iv), we get Solution Cont. This point must satisfy (i)

  15. Co-linearity of Three Points

  16. Example -5

  17. Angle Between Two Lines

  18. Cartesian Form

  19. Example –6

  20. Solution Cont.

  21. Example –7

  22. Solution Cont.

  23. Intersection of Two Lines Example - 8

  24. Solution Cont. If the lines intersect, then they have a common point.

  25. Solution Cont.

  26. P(1, 2, -3) Perpendicular Distance Example –9

  27. Solution Cont. Direction ratios of the given line are 2, - 2, -1.

  28. Solution Cont.

  29. P (2, -1, 5) L A B Q Image of a Point Example –10 Solution: Let Q be the image of the given point P(2, -1, 5) in the given line and let L be the foot of the perpendicular from P on the given line.

  30. Solution Cont.

  31. Solution Cont.

  32. Solution Cont. Hence, the image of P(2, -1, 5) in the given line is (0, 5, 1).

  33. Shortest Distance Between Two Lines Two straight lines in space, which do not intersect and are also not parallel, are called skew lines. (Which do not lie in the same plane)

  34. Cont.

  35. Cont.

  36. Cont.

  37. Example –11

  38. Solution Cont.

  39. Solution Cont.

  40. Example -12 Solution: The lines will intersect if shortest distance between them = 0 Therefore, the given lines intersect.

  41. Thank you

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