General Question

HeroicZach's avatar

How do you find the equation of the plane determined by two parameterized lines?

Asked by HeroicZach (195points) December 9th, 2010

I’m reviewing for my Calc II final tomorrow (ahh!) and my professor published a non-graded packet of review problems…with no answers. So yeah. He gives me two parameterized lines and asks me to find if they intersect and then find the equation for the plane that the two lines determine. Now, in order to check that they intersect, I took the cross product of the vectors implied by the parameterized line equations and saw that it was not zero – this means that the lines will intersect, because the vectors that define their direction are not parallel. However, I’m unsure how to approach finding the plane determined by these parameterized lines. I used the cross product I found as the normal to the plane, but I have two points on the plane to work with (the point you can pull out of each parameterized equation), and I’m not sure which one to put into the component form of the plane equation to determine the correct equation of the plane – I get two different equations depending on which of the points I use.

Any help you can give would be very appreciated!

Observing members: 0 Composing members: 0

3 Answers

Mariah's avatar

I think to write the equation of a plane, all you need is the equation of the normal to that plane (which you have) and any point on the plane (which you have). If I’m not mistaken, it doesn’t matter which of the points you use, because there are multiple equations that describe the same plane. Can that be right? I might have to go look through my notebook from last quarter and reply again, haha.

Mariah's avatar

Okay yeah I just checked my notebook.

The equation for a plane is in the form a(x-x0)+b(y-y0)+c(z-z0)=0 where <a, b, c> is the normal to the plane and <x0, y0, z0> is any point on the plane.

roundsquare's avatar

I could be wrong, but I don’t think you got the point of intersection correct. Just because the vectors along which the lines run aren’t parallel doesn’t mean the lines intersect. They could be skew.

E.g. the lines defined by (0, 0, u) and (1, v, 0) are not parallel but they never intersect.

To see if the two lines intersect, you need to match off the x, y & z coordinates.

So if your lines are (x1, y1, z1) and (x2, y2, z2) (where these are parametric functions) you solve:

x1(u) = x2(v)
y1(u) = y2(v)

This gets you a u and v. Plug those into z1 and z2 (respectively) and see if you get the same z coordinates. If so, they intersect. If not, they don’t.

If they intersect, call the point (xi, yi, zi). (i for point of intersection).

Now take the cross product and you get a vector (A, B, C).
So, take:
H = Axi + Byi + Czi

Whatever you get for H, your plane is:

Ax + By + Cz = H

Answer this question

Login

or

Join

to answer.

This question is in the General Section. Responses must be helpful and on-topic.

Your answer will be saved while you login or join.

Have a question? Ask Fluther!

What do you know more about?
or
Knowledge Networking @ Fluther