Finding the equation of a plane in 3-space

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In summary, you should be able to find the equation of the line of intersection between the two given planes by finding the points of intersection between the lines and the other two planes. Once you have done this, you can solve the rest of the problem exactly as before.
  • #1
HATE-VECTORS
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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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  • #2
If the point is not on the line, you should easily be able to create two vectors from the point to the line.
 
  • #3
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.
 
  • #4
And then? How do those two vectors help you find the plane?
 
  • #5
HATE-VECTORS said:
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?
 
  • #6
I know how it relates to finding the equation. My problem is that I have no clue how to find it:-)
 
  • #7
HATE-VECTORS said:
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?
 
  • #8
U multiply them together! and then that gives you the normal cause its perp to the plane as well! Thank you!
 
  • #9
ok. I get that now. What happens if Instead of being given the equation of a line I am 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 I am looking for?
 
  • #10
HATE-VECTORS said:
ok. I get that now. What happens if Instead of being given the equation of a line I am 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 I am 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.
 
  • #11
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??
 
  • #12
HATE-VECTORS said:
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??

You should be able to find the equation of the line of intersection between the two given planes. Once you have done this, you can solve the rest of the problem exactly as before, as long as the given point doesn't lie on the line.

If you're not sure how to find the line of intersection, then show us the given equations for the planes and where you got stuck.
 

FAQ: Finding the equation of a plane in 3-space

What is the equation of a plane in 3-space?

The equation of a plane in 3-space is a mathematical representation of a flat surface that extends infinitely in all directions. It is written in the form Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the variables x, y, and z respectively, and D is a constant term.

How do you find the equation of a plane in 3-space?

To find the equation of a plane in 3-space, you need to know either three non-collinear points on the plane or a normal vector and a point on the plane. Using this information, you can use the point-normal form or the three-point form to find the equation of the plane.

What is the point-normal form of the equation of a plane in 3-space?

The point-normal form of the equation of a plane in 3-space is written as (x, y, z) · n = (x0, y0, z0) · n, where (x, y, z) represents any point on the plane, n is the normal vector, and (x0, y0, z0) is a known point on the plane.

What is the three-point form of the equation of a plane in 3-space?

The three-point form of the equation of a plane in 3-space is written as (x - x0)(y1 - y0)(z2 - z0) - (x1 - x0)(y - y0)(z2 - z0) + (x1 - x0)(y2 - y0)(z - z0) = 0, where (x0, y0, z0), (x1, y1, z1), and (x2, y2, z2) are three non-collinear points on the plane.

What is the importance of finding the equation of a plane in 3-space?

Finding the equation of a plane in 3-space is important in many fields of science and engineering, such as physics, chemistry, and computer graphics. It allows us to describe and understand the behavior of objects and phenomena in three-dimensional space, and to make predictions and solve problems related to them.

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