1. MA242.003 Day 9 – January 17, 2013 Review: Equations of lines, Section 9.5 Section 9.5 –Planes

4. Equations of PLANES in space.

5. Different ways to specify a plane:

6. Equations of PLANES in space. 1. Give three non-co-linear points. Different ways to specify a plane:

7. Equations of PLANES in space. 1. Give three non-co-linear points. Different ways to specify a plane:

8. Equations of PLANES in space. 2. Give two non-parallel intersecting lines. Different ways to specify a plane:

9. Equations of PLANES in space. 2. Give two non-parallel intersecting lines. Different ways to specify a plane:

10. Equations of PLANES in space. Different ways to specify a plane: 3.Specify a point and a normal vector

11. Given:

12. Equations of PLANES in space.

14. Example: Find an equation for the plane containing the point (1,-5,2) with normal vector

15. Example: Find an equation for the plane containing the the points P=(1,-5,2), Q=(-3,8,2) and R=(0,-1,4)

16. REMARK: How equations of planes occur in problems

18. The Geometry of Lines and Planes For us, a LINE in space is a

19. The Geometry of Lines and Planes For us, a LINE in space is a Point and a direction vector v =

20. The Geometry of Lines and Planes For us, a Plane in space is a

21. The Geometry of Lines and Planes For us, a Plane in space is a Point on the plane And a normal vector n =

22. Two lines are parallel

23. when

24. Two lines are parallel when their direction vectors are parallel

25. Two lines are perpendicular

26. when

27. Two lines are perpendicular when their direction vectors are orthogonal

28. Two planes are parallel

29. when

30. Two planes are parallel when Their normal vectors are parallel

31. Two planes are perpendicular

32. when

33. Two planes are perpendicular when Their normal vectors are orthogonal

34. A line is parallel to a plane

35. when

36. A line is parallel to a plane when the direction vector v for the line is orthogonal to the normal vector n for the plane

37. A line is perpendicular to a plane

38. A line is perpendicular to a plane when

39. A line is perpendicular to a plane when the direction vector v for the line is parallel to the normal vector n for the plane

40. Example Problems