|
Intersection of Two Planes
Motivation:
What is the intersection of two planes ?
The two planes could be parallel to each other or they intersect. If they are not parallel to each other, what would be their intersection? Intuitively, the intersection should be a line. We are going to formulate this problem via equations.
Question ( Intersection of Two Lines ): Let E1 and E2 be two planes and the plane equations associated with them are E1 : x+y+z = 4 E2 : 2x-3y+z = -3
Find the equation of the intersection of the two planes.
Sol:
We have two equations with 3 variables x, y, and z; We are not going to have solution that x is equal to a number, y is equal to a number, and z is equal to a number. Conceptually, we know the solution is a “line equation”. Thus, there should exist many sets of numbers to satisfy the two equations.
Let’s try to solve in this way. Set z = t, and we have
x+y = 4-t 2x-3y= -3-t
x= – t y= - t
So, we have solution where t R
How do we know this is the solution we are looking for? We substitute them back to the two equations we have and we find out that this solution does satisfy the two equations we originally have. And the solution does look like a line equation.
We use another method to solve this problem.
Another method:
The normal vector of E1 is ( 1,1,1 ) , denoted as . The normal vector of E2 is ( 2, -3, 1) , denoted as .
We know that any vector lying on E1 is perpendicular to ; any vector lying on E2 is perpendicular to .
Thus, if a vector is lying on E1 and E2 , it is perpendicular to and ; In other words, the vector is parallel to .
= ( 4, -1, -5 )
1 1 1 1 1 2 -3 1 2 -3
can be used as the directional vector of the line; we also find out the point ( , , 0 ) is on both planes. So, the line equation can be
Actually, this set of equation is the same as we have by using the previous method if we set
The two solutions are the same. Of course, if we choose a different point on the line to use it in the equation, it will look different. But it still stands for the same line.
Question ( Angle between Two Planes ) Let E1 and E2 be two planes and the plane equations associated with them are E1 : x+y+z = 4 E2 : 2x-3y+z = -3
Find the angle between the two intersecting planes.
Sol: We still start from the normal vectors of the two planes; the angle between two planes should be the angle between the normal vectors of the two planes.
And let’s think further: there should be two angles of the two intersecting planes; the sum of the two angles is 1800 .
The two normal vectors are = ( 1, 1, 1 ) = ( 2, -3, 1 )
If the angle between them is denoted as , then
cos = = 0
Hence, we have = 900 . |
