Hi, I am not sure of the best strategy to approach finding the parametric description of an intersection line in 3-D where the two planes are also described parametrically.
The planes are described {t - u, 3t + u, u} and {4r - s, 0, s + r}
I know how to do this if the equations of the planes are Cartesian. Should I somehow convert them and eliminate the parameters?
Thanks!
Hello, cubs3205!
I've never had a plane described with two parameters.
But I followed my instincts and came up with a solution.
We have these two planes: . .(a)Hi, I am not sure of the best strategy to approach finding the parametric description
of an intersection line in 3-D where the two planes are also described parametrically.
The planes are described: {t - u, 3t + u, u} and {4r - s, 0, s + r}
Where they intersect, the 's, 's and 's are equal.
So we have: .
. . Add [1] and [3]: .
Then we have: .
. . Subtract [2] - [1]: .
Substitute into [3]: .
So we have: .
On the right, replace with a parameter
Substitute these into (a) and we have: .
And these are the parametric equations of the line of intersection.
For the first plane you have:
x = t - u ... (1)
y = 3t + u ... (2)
z = u ... (3)
Substitute (3) into (1) and (2):
x = t - z => t = x + z .... (A)
y = 3t + z .... (B)
Substitute (A) into (B):
y = 3(x + z) + z => 3x - y + 4z = 0.
This is the cartesian equation of the first plane.
The cartesian equation equation of the second plane is just y = 0.
So the line of intersection is found by solving 3x - y + 4z = 0 and y = 0 simultaneously: 3x + 4z = 0.
I'm sure you can express this line parametrically.
Edit: This is an alternative approach to Soroban (hey, you beat me this time! You must have started typing an hour ago lol!)