Need help finding equation of a plane

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In summary, the question is asking for the equation of a plane passing through a given point and parallel to a given line. The only known information is that the plane's direction vector is (1,-3,-1). To uniquely specify a plane, another piece of information is needed. The vector equation of a plane may be helpful in finding the solution.
  • #1
sk_88
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Homework Statement



Hi,

So my question is:

Find the equation of the plane which contains the point (0,1,3) and the line:

(x,y,z) = (-1,0,-2) + t(1,-3,-1)

It has to be in the form Ax+By+Cx=D. D has to be positive

Homework Equations



see above

The Attempt at a Solution



the only thing I know right now is that the plane has the direction vector of (1,-3,-1) because this plane is parallel to the line. And that could possibly be vital to finding the solution. But I have absolutely no idea where to go from there. So I hope you guys can help me out with this, thanks!
 
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  • #2
You know one direction vector, that is right.
This alone however is not enough to specify the plane you're lookng for.

What do you need to uniquely specify a plane?

For a line for example, two points would do.
 
  • #3
See Pere Callahan's Post.
Addition:
1) Don't go straight for Ax+By+Cz+D = 0
2) hint: you need to know what's the vector equation of a plane
 

What is the equation of a plane?

The equation of a plane is typically represented in the form of Ax + By + Cz = D, where A, B, and C are the coefficients of x, y, and z respectively, and D is a constant term.

How do I find the equation of a plane given three points?

To find the equation of a plane given three points, you can use the following steps:

  1. Calculate the vectors between two of the points: v1 = P2 - P1 and v2 = P3 - P1
  2. Find the cross product of the two vectors: v1 x v2 = (A, B, C)
  3. Use the values of A, B, and C to plug into the equation Ax + By + Cz = D
  4. Find the value of D by plugging in one of the points into the equation
  5. The resulting equation is the equation of the plane

Is there a different way to find the equation of a plane?

Another way to find the equation of a plane is by using a point on the plane and the normal vector of the plane. The equation can be written as (x - x0) * A + (y - y0) * B + (z - z0) * C = 0, where (x0, y0, z0) is the point and (A, B, C) is the normal vector.

What is a normal vector?

A normal vector is a vector that is perpendicular to a plane. It is used to determine the direction and orientation of the plane.

Can I find the equation of a plane in three-dimensional space?

Yes, the equation of a plane can be found in three-dimensional space using the same methods as in two-dimensional space. The only difference is that the equation will include three variables instead of two.

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