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 d r r0 y x Q’ P(x0,y0,z0) Q Q(x1,y1,z1) Q’(x,y,z)
r0=x0 i+y0 j+z0 k
d=d1 i+d2 j+d3 k r0
=(x1 -x0)i+(y1-y0)j+(z1-z0)k y x

6. Vector Parameterization
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+d3 k r0
=(x1 -x0)i+(y1-y0)j+(z1-z0)k y
Vector Parameterization x

7. Vector Parameterization
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+d3 k r0
=(x1 -x0)i+(y1-y0)j+(z1-z0)k y
Vector Parameterization x

8. Vector Parameterization
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+d3 k r0
=(x1 -x0)i+(y1-y0)j+(z1-z0)k y
Vector Parameterization x

9. Vector Parameterization
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+d3 k 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 y x 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 y Vector Equation x Scalar Equation 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 y Vector Equation x Scalar Equation 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 y Vector Equation x Scalar Equation 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 Planez R P X Q y P
Parametric Equation x

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

28. Parametric Equation of a Planez R P X 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. Example

35. 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.

36. Graphing Planes Sketch the following planes: (a) 3x - 2y + z = 6
(b) z + 2y = 4
(c) y = 2

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

38. Examples Find the equation of of the line through thez
Find the equation of
of the line through the
origin and perpendicular
to the plane pictured.
Find the equation of the
plane perpendicular to
x(t)=4-2t, y(t)= -1+t, z(t)=3 4 y 5 3 x

39. The Distance from a Point to a Plane

40. Distance from a Point to a Line
Let P0 be a point on l and let d be a direction vector for l. With P0 and Q as shown in the figure, you can see that