1. 11.5 Day 2 Lines and Planes in Space

2. Equations of planes

3. More equations of planes

4. The angle between two planes Two distinct planes in 3 dimensional space are parallel or intersect in a line. If they intersect, you can determine the angle (0 < ө < π) by finding the angle between a normal vector for each plane (see the diagram on the next slide) Finding the angle between two vectors is done by

5. Calculating the angle between two planes

6. Example 3 Find the general equation of the plane containing the points (2,1,1), (0,4,1) and (-2,1,4)

7. Example 4 Find the angle between the two planes and the line of intersection of the two planes x- 2y + z = 0 and 2x + 3y – 2z = 0

8. Solution to example 4a

9. Solution to example 4 b The line of intersection can be found be found by simultaneously solving the system of equations x - 2y + z = 0 multiply the top by -2 and add 2x + 3y - 2z = 0 yields 7y-4z =0 or y = 4z/7 Substitute this into the top equation you can get that x = z/7 set t = z/7 to obtain x = t, y = 4t, z =7t which is the parametric form of the line of intersection

11. Example 5 Find the distance between point Q (1,5,-4) And the plane given by 3x – y + 2z = 6

12. Solution to example 5

13. Example 6 Find the distance between the two parallel planes: 3x – y + 2z - 6 = 0 and 6x – 2y + 4z +4 = 0

14. Distance between two planes

15. Solution to example 6 To find the distance between the two planes first choose a point in the first plane say (2,0,0). Then from the second plane determine that a=6 b=-2 c=4 and d =4 (read these values directly from the second equation) The distance between the planes is given by:

17. Example 7 Find the distance between the point Q (3,-1,4) and the line given by x = -2 + 3t y = -2t z = 1+ 4t

19. "There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else -- but persistent." -- Raoul Bott “If I had only one day left to live, I would live it in my statistics class: it would seem so much longer.”