1. Lines in Space

2. z x y P Q Equation of a Line

3. z x y r0r0 d P Q

4. z x y r0r0 d r P Q Q’ Equation of a Line

5. z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

6. z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

7. z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

8. z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

9. z x y r0r0 d r P Q Q’ P(x 0,y 0,z 0 ) Q(x 1,y 1,z 1 ) Q’(x,y,z) Scalar Parametric Equations Vector Parameterization d=d 1 i+d 2 j+d 3 k r 0 =x 0 i+y 0 j+z 0 k =(x 1 -x 0 )i+(y 1 -y 0 )j+(z 1 -z 0 )k Equation of a Line

10. Representations of a Line

11. Examples

12. Planes in Space

13. z x y Equation of a Plane

14. z x y

15. z x y

16. z x y

17. z x y b

18. z x y c

19. z x y

20. z x y P n

21. z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Equation of a Plane

22. z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Scalar Equation Vector Equation Equation of a Plane

23. z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Scalar Equation Vector Equation Equation of a Plane

24. z x y Q P n r r Q Q(x,y,z) P(x 0,y 0,z 0 ) n=ai+bj+ck r=(x-x 0 )i+(y-y 0 )j+(z-z 0 )k Scalar Equation Vector Equation Equation of a Plane

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

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

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

29. Representations of a Plane Parametric Equation Scalar 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

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

36. Distance from a Point to a Line Let P 0 be a point on l and let d be a direction vector for l. With P 0 and Q as shown in the figure, you can see that

37. Application

38. The Distance from a Point to a Plane

39. Application