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Geometry in Space: Equations of Lines and Planes

Understand the equations of lines and planes in 3D space. Learn vector and scalar representations and solve applied problems with examples.

merritt-estes
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Geometry in Space: Equations of Lines and Planes

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  1. Lines in Space

  2. z Equation of a Line Q P y x

  3. z Equation of a Line Q d P r0 y x

  4. z Equation of a Line Q’ Q d P r r0 y x

  5. z Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y x

  6. z Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x

  7. z Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x

  8. z Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x

  9. z Equation of a Line Q’ P(x0,y0,z0) Q Q(x1,y1,z1) d Q’(x,y,z) P r r0=x0 i+y0 j+z0 k d=d1 i+d2 j+d3k r0 =(x1 -x0)i+(y1-y0)j+(z1-z0)k y Vector Parameterization x Scalar Parametric Equations

  10. Representations of a Line

  11. Examples

  12. Planes in Space

  13. z Equation of a Plane y x

  14. z Equation of a Plane y x

  15. z Equation of a Plane y x

  16. z Equation of a Plane y x

  17. z Equation of a Plane y b x

  18. z Equation of a Plane c y x

  19. z Equation of a Plane y x

  20. z Equation of a Plane n P y x

  21. z Equation of a Plane P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y x

  22. z Equation of a Plane P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y Vector Equation x Scalar Equation

  23. z Equation of a Plane P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y Vector Equation x Scalar Equation

  24. z Equation of a Plane P(x0,y0,z0) Q(x,y,z) n=ai+bj+ck r=(x-x0)i+(y-y0)j+(z-z0)k n Q r P r Q y Vector Equation x Scalar Equation

  25. Examples • Find the equation of the plane through (1,1,2), (3,2,-1) and (4,2,-1). • Find the equation of the plane through (2,-1,3) and parallel to 3x – y + 4z =12.

  26. Parametric Equation of a Plane z R X P Q y P Parametric Equation x

  27. Parametric Equation of a Plane z R X P Q y P Parametric Equation x

  28. Parametric Equation of a Plane z R X P Q y P Parametric Equation x

  29. Representations of a Plane Scalar Equation Parametric Equation

  30. Applications

  31. Angle Between Planes • Find the angle between the two planes 2x – 3y + 4z = 6 and x + 2y – 3z = -1

  32. Example

  33. Example

  34. Graphing Planes • Find the intercepts of the planes 2x – 3y + z = 6 4y + 2x = 8 z = 3 • Sketch the planes. • Find the normals to the planes.

  35. Examples z • Find the equation of the plane pictured. 4 y 5 3 x

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