1. 10.5 Lines and Planes in Space
Parametric Equations for a line in space
Linear equation for plane in space
Sketching planes given equations
Finding distance between points, planes, and lines in space

2. Sketching a plane

3. Use intercepts to find intersections with the coordinate axes (traces)

4. Equation of a line
vector value function, parametric equation, symmetric equation, standard form, and general form

5. Scenario 1: Line through a point, parallel to a vector

6. A line corresponds to the endpoints of a set of 2-dimensional position vectors.

7. Vector-valued function

8. Find a vector equation for the line that is parallel to the vector <0, 1, -3> and passes through the point <3, -2, 0>

9. Scenario 2: Line through 2 points

12. This gives the parametric equation of a line.
are the direction numbers of the line

13. Find the parametric equation of a line through the points
(2, -1, 5) and (7, -2, 3)

14. This gives the symmetric equation of a line.
Solving for t
This gives the symmetric equation of a line.
Write the line L through the point P = (2, 3, 5) and parallel to the vector
v=<4, -1, 6>, in the following forms:
Vector function
Parametric
Symmetric
Find two points on L distinct from P.

15. We can obtain an especially useful form of a line if we notice that
Substitute v into the equation for a line and reduce…

16. Intersection between two lines

18. Standard equation, general form, Functional form (*not in book)
Equation of a Plane
Standard equation, general form, Functional form (*not in book)

19. Scenario 1: normal vector and point
Given any plane, there must be at least one nonzero vector n = <a, b, c> that is perpendicular to every vector v parallel to the plane.

20. Standard Form or Point Normal Form
By regrouping terms, you obtain the general form of the equation of a plane:
ax+by+cz+d=0
(Standard form and general form are NOT unique!!!)
Solving for “z” will get you the functional form. (unique)

21. Find the equation of the plane with normal n = <1, 2, 7> which contains the point (5, 3, 4). Write in standard, general, and functional form.

22. Scenario 2: Three non-collinear points

23. Find the equation of the plane passing through
(1, 2, 2), (4, 6, 1), and (0, 5 4) in standard and functional form.
Note: using points in different order may result in a different normal and standard equation but the functional form will be the same.

24. Does it matter which point we use to plug into our standard equation?
Scenario 3: two lines
Does it matter which point we use to plug into our standard equation?

25. Scenario 4: Line and a point not on line
Find the equation of the plane containing the point (1, 2, 2) and the line L(t) = (4t+8, t+7, -3t-2)

27. Scenario 5: Span of two non-parallel vectors
Note: If u and v are parallel to a given plane P, then the plane P is said to be spanned by u and v.

28. Find the equation of the plane through the point (0, 0, 0) spanned by the vectors u= <1, 2, 1) and v = <3, 1, -2>

29. Intersection between 2 planes

30. Find the angle between the planes
Line:
Find the angle between the planes
x+2y-z=0 and x-y+3z+4=0

31. (a Write an equation for the line of intersection of the planes
x + y - z = 2 and 3x - 4y + 5z = 6
(b) find the angle between the planes.

32. Distance between points, planes, and lines

34. Given line L that goes through the points (-3, 1, -4) and (4, 4, -6), find the distance d from the point P = (1, 1, 1) to the line L.

37. Finding the distance between 2 parallel planes Ex. From pg. 758
Find the distance between the two parallel planes given by
3x-y+2z -6=0 and 6x-2y+4z+4=0

38. Finding the distance between 2 parallel planes
Find the distance between the two parallel planes given by
10x+2y-2z -6=0 and 5x+y-z-1=0

39. Homework:
Pg. 759/#1-7odd, 8, 9-13odd, 14-19, 21, 25-33odd, 37-51odd, 63, odd