How do I find the parametric equations for a plane?

  • Thread starter vg19
  • Start date
  • Tags
    Plane
In summary, to find the parametric equations for the plane 2x + y - z = 4, you can find the intercepts (2, 0, 0), (0, 4, 0), and (0, 0, -4) and create two direction vectors and a point. Alternatively, there may be an easier method, but it is currently unknown.
  • #1
vg19
67
0
Hi,

Given the plane 2x + y - z = 4, find its parametric equations.

This question seems simple, but the solution is not coming to me. I know the normal is (2,1,-1). But how do I find out the direction vectors and the points for the parametric equations?
 
Physics news on Phys.org
  • #2
I would suggest finding three points on the plane and create two direction vectors and a point for your parametric equation. The easiest three points would be the intercepts. if y and z are 0, the co-ordinate becomes: (2,0,0). Using a similar process, you can find the two other intercepts, which are:
(2, 0, 0)
(0, 4, 0)
(0, 0, -4)

Thus, you can create two direction vectors and a third point. I think there is an easier way to approach this, but for now it eludes me.
 
  • #3



Hi there,

To find the parametric equations for a plane, we need to first find two direction vectors that lie on the plane. One way to do this is by finding the cross product of the normal vector with any other vector that is not parallel to it. So in this case, we can choose the vector (1,0,0) to find the first direction vector. The cross product of (2,1,-1) and (1,0,0) is (0,-1,-2).

Next, we can find a second direction vector by taking the cross product of the first direction vector and the normal vector. So the second direction vector would be (-2,2,-1).

Now, to find the points for the parametric equations, we can choose any point on the plane and use it as the origin. Let's say we choose the point (0,0,4). Then, our parametric equations would be:

x = 0 + at + bs
y = 0 - ct
z = 4 + dt

Where a,b,c,d are any real numbers and t and s are the parameters that will help us generate points on the plane.

I hope this helps! Let me know if you have any other questions.
 

1. What is the equation of a plane?

The equation of a plane is a mathematical representation of a flat, two-dimensional surface in three-dimensional space. It is typically written in the form Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the variables x, y, and z, and D is a constant.

2. How do you find the equation of a plane?

To find the equation of a plane, you need to know at least three points that lie on the plane. From these points, you can determine the coefficients A, B, and C using a system of equations. Alternatively, you can also find the equation using a normal vector and a point on the plane.

3. Can the equation of a plane have fractions or decimals?

Yes, the coefficients of the variables in the equation of a plane can be fractions or decimals. This is because the equation represents a continuous surface and not just discrete points.

4. What is the significance of the normal vector in the equation of a plane?

The normal vector in the equation of a plane represents the direction perpendicular to the plane. It is important because it helps determine the orientation and angle of the plane in relation to other planes or objects in three-dimensional space.

5. Can the equation of a plane be used to solve real-world problems?

Yes, the equation of a plane has many practical applications in fields such as physics, engineering, and architecture. It can be used to describe the slope or angle of a surface, calculate distances and angles between objects, and determine the position of objects in space.

Similar threads

  • Introductory Physics Homework Help
Replies
33
Views
2K
  • Introductory Physics Homework Help
Replies
16
Views
339
  • Introductory Physics Homework Help
Replies
3
Views
697
  • Introductory Physics Homework Help
Replies
9
Views
1K
  • Introductory Physics Homework Help
Replies
4
Views
791
  • Introductory Physics Homework Help
Replies
10
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
880
  • Introductory Physics Homework Help
Replies
9
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
667
  • Introductory Physics Homework Help
Replies
3
Views
696
Back
Top