# Find equation of plane given conditions

• Mr Davis 97
In summary, the conversation discusses finding an equation of a plane that passes through a given point and contains a given line. The equation of a plane is determined by a base vector and a normal vector. The suggested approach is to find two vectors in the plane using the given line, and then use the given point and these vectors to find the equation of the plane.
Mr Davis 97

## Homework Statement

Find an equation of the plane that passes through the point (1,2,-2) and that contains the line
x=2t,y=3-t,z=1+3t.

## The Attempt at a Solution

I know that a plane is determined by a base vector and a normal vector, and the equation of the plane is ##\vec{n} \cdot (\vec{r} - \vec{r_0})##, where n is the normal vector, and r0 is the base vector.

So we know the base vector, and now must determine what the normal vector is to the plane. We need to find two vectors in the plane, and take the cross product of those vectors. I am not sure where to find those two vectors... Would any two non-parallel vectors whose endpoints are on the line suffice?

Hi Mr Davis:
I suggest thinking about the problem a bit differently.

I think it would help if you write down, in section 2, the general form of the "equation of the plane". The identity of three points in the plane should lead you to three simultaneous equations to solve.

Good luck.

Regards,
Buzz

Is the given point on the line? If so, there is no unique plane.

If the given point is not on the line, use the parametric equations of the line to find two points. then you'll have three points, from which you can get two vectors in the plane. From these vectors, you can get a normal to the plane, and from it and the given point, getting the equation of the plane is straightforward.

Mr Davis 97

## What is the equation of a plane?

The equation of a plane is a mathematical expression that represents a flat, two-dimensional surface in three-dimensional space. It is typically written in the form ax + by + cz = d, where a, b, and c are the coefficients and x, y, and z are the variables.

## How do you find the equation of a plane given conditions?

To find the equation of a plane, you need to know at least three points that lie on the plane or two vectors that are parallel to the plane. From this information, you can use the point-normal form of the plane equation or the cross product of the two vectors to determine the coefficients of the equation.

## What is the point-normal form of a plane equation?

The point-normal form of a plane equation is (x - x0) · n = 0, where x0 is a point on the plane and n is the normal vector to the plane. This form is useful for finding the equation of a plane when you know a point on the plane and the normal vector to the plane.

## Can you find the equation of a plane with only two points?

No, you need at least three points or two vectors to determine the equation of a plane. This is because there are an infinite number of planes that can contain two points, and you need the third point or vector to uniquely define the plane.

## What is the significance of the coefficients in the plane equation?

The coefficients in the plane equation represent the direction and orientation of the plane in relation to the x, y, and z axes. The values of a, b, and c determine the slope of the plane in the x, y, and z directions, respectively, while the value of d represents the distance of the plane from the origin.

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