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. VECTOR VALUE FUNCTION, PARAMETRIC EQUATION, SYMMETRIC EQUATION, STANDARD FORM, AND GENERAL FORM Equation of a line

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 and passes through the point

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

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

16. INTERSECTION BETWEEN TWO LINES

18. STANDARD EQUATION, GENERAL FORM, FUNCTIONAL FORM (*NOT IN BOOK) Equation of a Plane

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

20. 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) Standard Form or Point Normal Form

21. Find the equation of the plane with normal n = 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. 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. INTERSECTION BETWEEN 2 PLANES

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

29. 1a. Write an equation for the line of intersection of the planes x + y - z = 2 and 3x - 4y + 5z = 6 1b. find the angle between the planes. 2a. Write an equation for the line of intersection of the planes 5x-3y+z-10=0 and 2x+4y-x+3=0 2b. Find the angle between the planes Example of parallel planes will be in a future slide---for those problems, we only find the distance between the planes. Examples of intersections of planes (note: these are not scalar multiples of each other…therefore NOT parallel!

30. DISTANCE BETWEEN POINTS, PLANES, AND LINES

31. *This formula is from your cross product and sine formula.

32. 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.

35. 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

36. 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

37. PG. 759/#1-7ODD, 8, 9-13ODD, 14-19, 21, 25-33ODD, 37-51ODD, 63, 67-81 ODD Homework: