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)

8. 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

9. Solution to example 4a

10. 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

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

13. Solution to example 5

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

15. Distance between two planes

16. 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:

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

20. "There are two ways to do great mathematics
"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.”