Find the Intersection of Two Planes Given in Parametric Form

Understanding Parametric Form

To determine the intersection of two planes, it's often convenient to have them represented in parametric form. Here's an explanation of parametric form:

• Plane 1: r₁ = a₁ + t₁b₁ + s₁c₁
• Plane 2: r₂ = a₂ + t₂b₂ + s₂c₂

where: * r₁, r₂ represent the position vectors in the first and second planes, respectively * a₁, a₂ are fixed points in the first and second planes, respectively * b₁, b₂ are vectors parallel to the first plane * c₁, c₂ are vectors parallel to the second plane * t₁, t₂ are scalar parameters * s₁, s₂ are scalar parameters

Determining Intersection Using Equations

To find the intersection point, we set r₁ = r₂ and solve for the parameters t₁, t₂, s₁, and s₂. This results in a system of equations that can be solved using techniques like substitution, elimination, or matrix methods.

The solution to the system of equations will give us the values of t₁, t₂, s₁, and s₂ that correspond to the intersection point. Once we have these values, we can substitute them back into either plane equation to find the coordinates of the intersection point.

Advantages of Parametric Form

Representing planes in parametric form offers several advantages:

• Compact representation: Parametric form allows for a concise representation of planes, especially when they are parallel or intersecting.
• Simplicity: Solving for the intersection of planes in parametric form is often more straightforward than using other forms like Cartesian or vector forms.
• Generalization: Parametric form can be extended to higher dimensions, making it applicable to various geometric problems.

Conclusion

Understanding and utilizing parametric form is a valuable tool for finding the intersection of two planes. By converting planes to parametric equations and solving a system of equations, we can efficiently determine the intersection point and its coordinates.