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- 2y + z = 0 and 2x + 3y – 2z = 0

Solution to example 4a

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

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

Solution to example 5

Example 6 Find the distance between the two parallel planes: 3x – y + 2z - 6 = 0 and 6x – 2y + 4z +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 + 3t y = -2t z = 1+ 4t

"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.”