11 5 Day 2 Lines and Planes in
- Slides: 19
11. 5 Day 2 Lines and Planes in Space
Equations of planes
More equations of planes
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
Calculating the angle between two planes
Example 3 • Find the general equation of the plane containing the points (2, 1, 1), (0, 4, 1) and (-2, 1, 4)
Example 4 Find the angle between the two planes and the line of intersection of the two planes x- 2 y + z = 0 and 2 x + 3 y – 2 z = 0
Solution to example 4 a
Solution to example 4 b The line of intersection can be found by simultaneously solving the system of equations x - 2 y + z = 0 multiply the top by -2 and add 2 x + 3 y - 2 z = 0 yields 7 y-4 z =0 or y = 4 z/7 Substitute this into the top equation you can get that x = z/7 set t = z/7 to obtain x = t, y = 4 t, z =7 t which is the parametric form of the line of intersection
Example 5 • Find the distance between point Q (1, 5, -4) And the plane given by 3 x – y + 2 z = 6
Solution to example 5
Example 6 Find the distance between the two parallel planes: 3 x – y + 2 z - 6 = 0 and 6 x – 2 y + 4 z +4 = 0
Distance between two planes
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:
Example 7 Find the distance between the point Q (3, -1, 4) and the line given by x = -2 + 3 t y = -2 t z = 1+ 4 t
"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. ”
