Determine the Intersection of Two Planes with Ease

Calculating Intersecting Points

Understanding the intersection of two planes is crucial for comprehending their spatial relationship. When one plane is given in coordinate form and the other in parametric form, the point of intersection can be calculated precisely.

Determining the Intersection Point

To determine the intersection point, we need to find the point that lies on both planes simultaneously. This point can be represented parametrically as:

``` x = x₀ + r * a y = y₀ + r * b z = z₀ + r * c ```

where (x₀, y₀, z₀) is a point on the coordinate plane and (a, b, c) is the direction vector of the parametric plane.

Substituting these equations into the coordinate plane equation, we obtain a system of three linear equations in terms of the parameter r:

``` (a₀ - x₀) + r * (a - a₀) = 0 (b₀ - y₀) + r * (b - b₀) = 0 (c₀ - z₀) + r * (c - c₀) = 0 ```

Solving this system gives us the value of r, which we can then substitute back into the parametric equations to find the coordinates of the intersection point.

Simplifying the Calculation

To simplify the calculations further, we can use a matrix method to solve for r:

``` [a - a₀, b - b₀, c - c₀] * [r] = [x₀ - a₀, y₀ - b₀, z₀ - c₀] ```

Solving for r using Cramer's rule or Gauss-Jordan elimination will give us the value of the parameter that corresponds to the intersection point.