determine the equation of a plane, P3, that intersects the planes P1: x + y + 3z − 2 = 0 and P2: x − y + 2z = 0 in
a point;
a line.
I know the normals are (1,1,3) and (1,-1,2) clearly the lines don't intersect because the normals are't multiples. So I don't know how to start the question
well, any old plane that is not parallel to the other two will intersect both at a point along the lines of intersection of those planes.
The line of intersection will be parallel to the cross product of or two normals
i j k
1 1 3
1-1 2
Find a plane that has that line of intersection as a normal and you have one that hits both planes at a point on their intersection.
Now to intersect both in the same line, your plane better include the line of intersection of the first two. Any old plane that includes that line will do (except either of the original two of course :)
To determine the equation of a plane, P3, that intersects both P1 and P2, we can start by finding a point that lies on both planes. Since the normals of P1 and P2 are not multiples of each other, they don't intersect in a line.
1. Finding a point of intersection:
To find a point that lies on both planes, we can solve the system of equations formed by the equations of P1 and P2:
P1: x + y + 3z - 2 = 0
P2: x - y + 2z = 0
Let's solve this system of equations using an elimination method:
Multiply equation P2 by 3:
3(x - y + 2z) = 0
3x - 3y + 6z = 0
Now subtract P1 from this new equation:
3x - 3y + 6z - (x + y + 3z - 2) = 0
2x - 4y + 3z + 2 = 0
Simplify:
2x - 4y + 3z = -2
2x/(-2) - 4y/(-2) + 3z/(-2) = -2/(-2)
-x + 2y - (3/2)z = 1
We now have a new plane, P3: -x + 2y - (3/2)z = 1, which intersects both P1 and P2 at a point.
2. Finding a line of intersection:
To find a line of intersection between P1 and P2, we can use the cross product of their normals. The vector formed by the cross product will be the direction vector of the intersecting line.
Cross product of the normals (1,1,3) and (1,-1,2):
i j k
1 1 3
1 -1 2
= (5, 1, -2)
The direction vector of the line of intersection is (5, 1, -2).
Thus, the equation of the plane, P3, that intersects P1 and P2 is:
- x + 2y - (3/2)z = 1
And the line of intersection between P1 and P2 has the direction vector (5, 1, -2).
To determine the equation of a plane, P3, that intersects two planes, P1 and P2, in different ways, we need to consider the different possible cases.
1. Intersection in a point:
If the planes P1 and P2 intersect in a point, it means their normals are not parallel. In this case, the equation of the plane P3 passing through the common point can be found by taking the cross product of the normals of P1 and P2.
The equation of a plane can be written in the form: ax + by + cz + d = 0, where (a, b, c) is the normal to the plane. To find the normal, we can take the cross product of the given normals.
Given: Normal of P1 = (1, 1, 3) and Normal of P2 = (1, -1, 2)
Calculating the cross product, we have:
Normal of P3 = (1, 1, 3) × (1, -1, 2) = (-5, -1, -2)
Therefore, the equation of plane P3 passing through the common point of intersection can be written as:
-5x - y - 2z + d = 0
Now, to find the value of "d" (the constant term in the equation), substitute the coordinates of the common point of intersection into the equation of P3. This will give you the complete equation of plane P3.
2. Intersection in a line:
If the normals of P1 and P2 are parallel or multiples of each other, it means the planes P1 and P2 are parallel or coincide. In this case, there is an infinite number of planes that can intersect both P1 and P2 in a line.
To find the equation of such a plane, you can use the direction vector of the line of intersection. To obtain this direction vector, you can take the cross product of the normals of P1 and P2.
Continuing from the given information:
Direction vector of the line of intersection = (1, 1, 3) × (1, -1, 2) = (-5, -1, -2)
The equation of the plane P3 passing through the line of intersection can be written as:
-5x - y - 2z + d = 0
Again, to determine the value of "d," substitute the coordinates of any point on the line of intersection into the equation of P3.
Please note that the value of "d" will vary depending on the choice of the point on the line of intersection.