Calculate Line Where Plane Intersects 0

What is plane intersection with 0?

When a plane intersects the origin (0,0,0) in 3D space, it means the plane passes through this point. The intersection line is the set of all points that lie on both the plane and pass through the origin.

This concept is important in geometry, physics, and computer graphics where understanding how surfaces intersect is crucial for modeling and simulation.

Formula for the intersection line

A plane in 3D space can be defined by the equation:

where (a, b, c) is the normal vector to the plane, and d is a constant.

If the plane passes through the origin (0,0,0), then substituting x=0, y=0, z=0 gives:

Therefore, the plane equation simplifies to:

The line of intersection is any line that lies entirely on this plane and passes through the origin. A parametric equation for this line can be written as:

where t is a parameter that can take any real value, and (b·c, -a·c, a·b) is a direction vector perpendicular to the normal vector (a, b, c).

How to use this calculator

1. Enter the coefficients a, b, and c of the plane equation in the calculator.
2. Click "Calculate" to find the parametric equation of the intersection line.
3. The result will show the direction vector and the parametric equation.
4. Use the chart to visualize the line in 3D space.

Worked example

Let's find the line where the plane x + 2y + 3z = 0 intersects the origin.

Here, a=1, b=2, c=3. The direction vector is (b·c, -a·c, a·b) = (6, -3, 2).

The parametric equation of the line is:

This means for any value of t, the point (6t, -3t, 2t) lies on both the plane and passes through the origin.