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Equations of Planes in Space: Understanding Specifications and Geometry

This lesson explores the equations of planes in three-dimensional space. It covers diverse methods for specifying a plane, including the use of three non-collinear points, two non-parallel intersecting lines, and a point with a normal vector. Examples illustrate how to find the equation of a plane given a point and a normal vector, as well as using three points in space. Additionally, the relationship between lines and planes, such as conditions for parallelism and perpendicularity, is discussed, enhancing the understanding of geometric relationships.

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Equations of Planes in Space: Understanding Specifications and Geometry

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

  2. Equations of PLANES in space.

  3. Equations of PLANES in space. Different ways to specify a plane:

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

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

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

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

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

  9. Given:

  10. Equations of PLANES in space.

  11. Equations of PLANES in space.

  12. Example: Find an equation for the plane containing the point (1,-5,2) with normal vector <-3,7,5>

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

  14. REMARK: How equations of planes occur in problems

  15. REMARK: How equations of planes occur in problems

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

  17. The Geometry of Lines and Planes • For us, a LINE in space is a Point and a direction vector v = <a,b,c>

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

  19. The Geometry of Lines and Planes • For us, a Plane in space is a Point on the plane And a normal vector n = <a,b,c>

  20. Two lines are parallel

  21. Two lines are parallel when

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

  23. Two lines are perpendicular

  24. Two lines are perpendicular when

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

  26. Two planes are parallel

  27. Two planes are parallel when

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

  29. Two planes are perpendicular

  30. Two planes are perpendicular when

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

  32. A line is parallel to a plane

  33. A line is parallel to a plane when

  34. 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

  35. A line is perpendicular to a plane

  36. A line is perpendicular to a plane when

  37. 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

  38. Example Problems

  39. Example Problems

  40. Example Problems

  41. Example Problems

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