# Finding the equation of a plane in 3-space

## Main Question or Discussion Point

How do you find an equation of a plane in 3-space when you are only given one point and the parametric equation of a line which is contained within the plane? I tried making an augmented matrix but only have 2 equations for 3 unknowns!!!

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If the point is not on the line, you should easily be able to create two vectors from the point to the line.

HallsofIvy
Homework Helper
Yes. If the point is on the line, then there are an infinite number of such planes.

If the point is not on the line, choose any two points on the line and construct the vectors from each to the given point.

And then? How do those two vectors help you find the plane?

jbunniii
Homework Helper
Gold Member
And then? How do those two vectors help you find the plane?
Do you know how to find a normal vector, and how it relates to finding an equation for the plane?

I know how it relates to finding the equation. My problem is that I have no clue how to find it:-)

jbunniii
Homework Helper
Gold Member
I know how it relates to finding the equation. My problem is that I have no clue how to find it:-)
If you have two vectors that are not parallel, do you know an operation that always produces a third vector that is perpendicular to both of them?

U multiply them together!!! and then that gives you the normal cause its perp to the plane as well!!! Thank you!!!

ok. I get that now. What happens if Instead of being given the equation of a line im told that the plane contains a line that intersects two other planes. And then I am given just the equation of the two other planes and a point on the plane that im looking for?

jbunniii
Homework Helper
Gold Member
ok. I get that now. What happens if Instead of being given the equation of a line im told that the plane contains a line that intersects two other planes. And then I am given just the equation of the two other planes and a point on the plane that im looking for?
It depends on what information is given. Do you have enough information to determine the two points of intersection between the line and the other two planes?

In general, all you need are three points in the plane of interest, where not all of them are on the same line. Once you have them, you can proceed as before.

I am not given the equation of a line at all. I am told that two planes (of which the equations are given) intersect at a line. That line passes through another plane. I am given a point in the plane and told to find it's equation> now from what i understand i can't do much with the equations of two planes cause there are three unknown... unl;ess- am i ment to solve in terms of the unkown and use that to choose any two random points and then make vectors to the given point and do what you just told me to do??

jbunniii