Lesson 4: Lines and Planes (slides + notes)

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Lesson 4: Lines and Planes (slides + notes)

  1. 1. Lesson 4 Lines and Planes Math 20 September 26, 2007 Announcements Problem Set 1 is due today Problem Set 2 is on the course web site. Due October 3 My office hours: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
  2. 2. Lines in the plane There are many ways to specify a line in the plane:
  3. 3. Lines in the plane There are many ways to specify a line in the plane: two points point and slope slope and intercept
  4. 4. Lines in the plane There are many ways to specify a line in the plane: two points point and slope slope and intercept How can we specify a line in three or more dimensions?
  5. 5. Using vectors to describe lines Let y = mx + b be a line in the plane. Let
  6. 6. Using vectors to describe lines Let y = mx + b be a line in the plane. Let 0 a= b a
  7. 7. Using vectors to describe lines Let y = mx + b be a line in the plane. Let v 0 1 a= v= b m a
  8. 8. Using vectors to describe lines Let y = mx + b be a line in the plane. Let v 0 1 a= v= b m a Then the line can be described as the set of all 0 1 t x = a + tv = +t = b m mt + b as t ranges over all real numbers.
  9. 9. Generalizing Any line in Rn can be described by a point a and a direction v and given parametrically by the equation x = a + tv
  10. 10. Applying the definition Example Determine if the points a = (1, 2, 3), b = (3, 5, 7), and c = (4, 8, 11) in R3 are on the same line.
  11. 11. Applying the definition Example Determine if the points a = (1, 2, 3), b = (3, 5, 7), and c = (4, 8, 11) in R3 are on the same line. Solution They are on the same line if c is on the line specified by a and b. So we will find the equation for this line and test if c is on it. The line has a on it and goes in the direction b − a. So it can be written in the form       1 2 1 + 2t x = 2 + t 3 = 2 + 3t  3 4 3 + 4t
  12. 12. Solution (continued) c is on this line if this system of equations has a solution: 1 + 2t = 5 2 + 3t = 8 3 + 4t = 11 The first one tells us t = 3/2, but the second t = 2. So there is no solution of all three.
  13. 13. Generalizing Any line in Rn can be described by a point a and a direction v and given parametrically by the equation x = a + tv Alternatively, any line in Rn can be described by two points a and b by letting a be the point and b − a the direction.
  14. 14. Generalizing Any line in Rn can be described by a point a and a direction v and given parametrically by the equation x = a + tv Alternatively, any line in Rn can be described by two points a and b by letting a be the point and b − a the direction. Then x = a + t(b − a) = (1 − t)a + tb.
  15. 15. Lines in the plane, again Let p be perpendicular to v. p v a
  16. 16. Lines in the plane, again a Let p be perpendicular to v. x− Then the head of x is on the p v line exactly when x − a is x parallel to v, or perpendicular a to p.
  17. 17. Lines in the plane, again a Let p be perpendicular to v. x− Then the head of x is on the p v line exactly when x − a is x parallel to v, or perpendicular a to p. So the line can be described as the set of all x such that p · (x − a) = 0
  18. 18. Generalizing again This generalizes to R3 as well. z p a y x
  19. 19. Generalizing again This generalizes to R3 as well. z p a y x
  20. 20. Generalizing again This generalizes to R3 as well. z p a y x This time, the “locus” is a plane.
  21. 21. Example Find the equation of the plane that passes through the points (1, 2, 3), (3, 5, 7), and (4, 3, 1)
  22. 22. Hyperplanes in Rn Definition A hyperplane through a that is orthogonal to a vector p = 0 is the set of all points x satisfying p · (x − a) = 0.

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