11.5 Day 2 Lines and Planes in Space
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
Example 3
• Find the general equation of the plane containing the points (2,1,1), (0,4,1) and
(-2,1,4)
Example 4Find 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 4 bThe 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 6
Find the distance between the two parallel planes: 3x – y + 2z - 6 = 0 and
6x – 2y + 4z +4 = 0
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